This file is indexed.

/usr/share/gap/doc/ref/chap39.html is in gap-doc 4r8p6-2.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

   1
   2
   3
   4
   5
   6
   7
   8
   9
  10
  11
  12
  13
  14
  15
  16
  17
  18
  19
  20
  21
  22
  23
  24
  25
  26
  27
  28
  29
  30
  31
  32
  33
  34
  35
  36
  37
  38
  39
  40
  41
  42
  43
  44
  45
  46
  47
  48
  49
  50
  51
  52
  53
  54
  55
  56
  57
  58
  59
  60
  61
  62
  63
  64
  65
  66
  67
  68
  69
  70
  71
  72
  73
  74
  75
  76
  77
  78
  79
  80
  81
  82
  83
  84
  85
  86
  87
  88
  89
  90
  91
  92
  93
  94
  95
  96
  97
  98
  99
 100
 101
 102
 103
 104
 105
 106
 107
 108
 109
 110
 111
 112
 113
 114
 115
 116
 117
 118
 119
 120
 121
 122
 123
 124
 125
 126
 127
 128
 129
 130
 131
 132
 133
 134
 135
 136
 137
 138
 139
 140
 141
 142
 143
 144
 145
 146
 147
 148
 149
 150
 151
 152
 153
 154
 155
 156
 157
 158
 159
 160
 161
 162
 163
 164
 165
 166
 167
 168
 169
 170
 171
 172
 173
 174
 175
 176
 177
 178
 179
 180
 181
 182
 183
 184
 185
 186
 187
 188
 189
 190
 191
 192
 193
 194
 195
 196
 197
 198
 199
 200
 201
 202
 203
 204
 205
 206
 207
 208
 209
 210
 211
 212
 213
 214
 215
 216
 217
 218
 219
 220
 221
 222
 223
 224
 225
 226
 227
 228
 229
 230
 231
 232
 233
 234
 235
 236
 237
 238
 239
 240
 241
 242
 243
 244
 245
 246
 247
 248
 249
 250
 251
 252
 253
 254
 255
 256
 257
 258
 259
 260
 261
 262
 263
 264
 265
 266
 267
 268
 269
 270
 271
 272
 273
 274
 275
 276
 277
 278
 279
 280
 281
 282
 283
 284
 285
 286
 287
 288
 289
 290
 291
 292
 293
 294
 295
 296
 297
 298
 299
 300
 301
 302
 303
 304
 305
 306
 307
 308
 309
 310
 311
 312
 313
 314
 315
 316
 317
 318
 319
 320
 321
 322
 323
 324
 325
 326
 327
 328
 329
 330
 331
 332
 333
 334
 335
 336
 337
 338
 339
 340
 341
 342
 343
 344
 345
 346
 347
 348
 349
 350
 351
 352
 353
 354
 355
 356
 357
 358
 359
 360
 361
 362
 363
 364
 365
 366
 367
 368
 369
 370
 371
 372
 373
 374
 375
 376
 377
 378
 379
 380
 381
 382
 383
 384
 385
 386
 387
 388
 389
 390
 391
 392
 393
 394
 395
 396
 397
 398
 399
 400
 401
 402
 403
 404
 405
 406
 407
 408
 409
 410
 411
 412
 413
 414
 415
 416
 417
 418
 419
 420
 421
 422
 423
 424
 425
 426
 427
 428
 429
 430
 431
 432
 433
 434
 435
 436
 437
 438
 439
 440
 441
 442
 443
 444
 445
 446
 447
 448
 449
 450
 451
 452
 453
 454
 455
 456
 457
 458
 459
 460
 461
 462
 463
 464
 465
 466
 467
 468
 469
 470
 471
 472
 473
 474
 475
 476
 477
 478
 479
 480
 481
 482
 483
 484
 485
 486
 487
 488
 489
 490
 491
 492
 493
 494
 495
 496
 497
 498
 499
 500
 501
 502
 503
 504
 505
 506
 507
 508
 509
 510
 511
 512
 513
 514
 515
 516
 517
 518
 519
 520
 521
 522
 523
 524
 525
 526
 527
 528
 529
 530
 531
 532
 533
 534
 535
 536
 537
 538
 539
 540
 541
 542
 543
 544
 545
 546
 547
 548
 549
 550
 551
 552
 553
 554
 555
 556
 557
 558
 559
 560
 561
 562
 563
 564
 565
 566
 567
 568
 569
 570
 571
 572
 573
 574
 575
 576
 577
 578
 579
 580
 581
 582
 583
 584
 585
 586
 587
 588
 589
 590
 591
 592
 593
 594
 595
 596
 597
 598
 599
 600
 601
 602
 603
 604
 605
 606
 607
 608
 609
 610
 611
 612
 613
 614
 615
 616
 617
 618
 619
 620
 621
 622
 623
 624
 625
 626
 627
 628
 629
 630
 631
 632
 633
 634
 635
 636
 637
 638
 639
 640
 641
 642
 643
 644
 645
 646
 647
 648
 649
 650
 651
 652
 653
 654
 655
 656
 657
 658
 659
 660
 661
 662
 663
 664
 665
 666
 667
 668
 669
 670
 671
 672
 673
 674
 675
 676
 677
 678
 679
 680
 681
 682
 683
 684
 685
 686
 687
 688
 689
 690
 691
 692
 693
 694
 695
 696
 697
 698
 699
 700
 701
 702
 703
 704
 705
 706
 707
 708
 709
 710
 711
 712
 713
 714
 715
 716
 717
 718
 719
 720
 721
 722
 723
 724
 725
 726
 727
 728
 729
 730
 731
 732
 733
 734
 735
 736
 737
 738
 739
 740
 741
 742
 743
 744
 745
 746
 747
 748
 749
 750
 751
 752
 753
 754
 755
 756
 757
 758
 759
 760
 761
 762
 763
 764
 765
 766
 767
 768
 769
 770
 771
 772
 773
 774
 775
 776
 777
 778
 779
 780
 781
 782
 783
 784
 785
 786
 787
 788
 789
 790
 791
 792
 793
 794
 795
 796
 797
 798
 799
 800
 801
 802
 803
 804
 805
 806
 807
 808
 809
 810
 811
 812
 813
 814
 815
 816
 817
 818
 819
 820
 821
 822
 823
 824
 825
 826
 827
 828
 829
 830
 831
 832
 833
 834
 835
 836
 837
 838
 839
 840
 841
 842
 843
 844
 845
 846
 847
 848
 849
 850
 851
 852
 853
 854
 855
 856
 857
 858
 859
 860
 861
 862
 863
 864
 865
 866
 867
 868
 869
 870
 871
 872
 873
 874
 875
 876
 877
 878
 879
 880
 881
 882
 883
 884
 885
 886
 887
 888
 889
 890
 891
 892
 893
 894
 895
 896
 897
 898
 899
 900
 901
 902
 903
 904
 905
 906
 907
 908
 909
 910
 911
 912
 913
 914
 915
 916
 917
 918
 919
 920
 921
 922
 923
 924
 925
 926
 927
 928
 929
 930
 931
 932
 933
 934
 935
 936
 937
 938
 939
 940
 941
 942
 943
 944
 945
 946
 947
 948
 949
 950
 951
 952
 953
 954
 955
 956
 957
 958
 959
 960
 961
 962
 963
 964
 965
 966
 967
 968
 969
 970
 971
 972
 973
 974
 975
 976
 977
 978
 979
 980
 981
 982
 983
 984
 985
 986
 987
 988
 989
 990
 991
 992
 993
 994
 995
 996
 997
 998
 999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
2873
2874
2875
2876
2877
2878
2879
2880
2881
2882
2883
2884
2885
2886
2887
2888
2889
2890
2891
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908
2909
2910
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
3037
3038
3039
3040
3041
3042
3043
3044
3045
3046
3047
3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
3077
3078
3079
3080
3081
3082
3083
3084
3085
3086
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
3108
3109
3110
3111
3112
3113
3114
3115
3116
3117
3118
3119
3120
3121
3122
3123
3124
3125
3126
3127
3128
3129
3130
3131
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161
3162
3163
3164
3165
3166
3167
3168
3169
3170
3171
3172
3173
3174
3175
3176
3177
3178
3179
3180
3181
3182
3183
3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
3204
3205
3206
3207
3208
3209
3210
3211
3212
3213
3214
3215
3216
3217
3218
3219
3220
3221
3222
3223
3224
3225
3226
3227
3228
3229
3230
3231
3232
3233
3234
3235
3236
3237
3238
3239
3240
3241
3242
3243
3244
3245
3246
3247
3248
3249
3250
3251
3252
3253
3254
3255
3256
3257
3258
3259
3260
3261
3262
3263
3264
3265
3266
3267
3268
3269
3270
3271
3272
3273
3274
3275
3276
3277
3278
3279
3280
3281
3282
3283
3284
3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
3300
3301
3302
3303
3304
3305
3306
3307
3308
3309
3310
3311
3312
3313
3314
3315
3316
3317
3318
3319
3320
3321
3322
3323
3324
3325
3326
3327
3328
3329
3330
3331
3332
3333
3334
3335
3336
3337
3338
3339
3340
3341
3342
3343
3344
3345
3346
3347
3348
3349
3350
3351
3352
3353
3354
3355
3356
3357
3358
3359
3360
3361
3362
3363
3364
3365
3366
3367
3368
3369
3370
3371
3372
3373
3374
3375
3376
3377
3378
3379
3380
3381
3382
<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (ref) - Chapter 39: Groups</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap39"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chap16.html">16</a>  <a href="chap17.html">17</a>  <a href="chap18.html">18</a>  <a href="chap19.html">19</a>  <a href="chap20.html">20</a>  <a href="chap21.html">21</a>  <a href="chap22.html">22</a>  <a href="chap23.html">23</a>  <a href="chap24.html">24</a>  <a href="chap25.html">25</a>  <a href="chap26.html">26</a>  <a href="chap27.html">27</a>  <a href="chap28.html">28</a>  <a href="chap29.html">29</a>  <a href="chap30.html">30</a>  <a href="chap31.html">31</a>  <a href="chap32.html">32</a>  <a href="chap33.html">33</a>  <a href="chap34.html">34</a>  <a href="chap35.html">35</a>  <a href="chap36.html">36</a>  <a href="chap37.html">37</a>  <a href="chap38.html">38</a>  <a href="chap39.html">39</a>  <a href="chap40.html">40</a>  <a href="chap41.html">41</a>  <a href="chap42.html">42</a>  <a href="chap43.html">43</a>  <a href="chap44.html">44</a>  <a href="chap45.html">45</a>  <a href="chap46.html">46</a>  <a href="chap47.html">47</a>  <a href="chap48.html">48</a>  <a href="chap49.html">49</a>  <a href="chap50.html">50</a>  <a href="chap51.html">51</a>  <a href="chap52.html">52</a>  <a href="chap53.html">53</a>  <a href="chap54.html">54</a>  <a href="chap55.html">55</a>  <a href="chap56.html">56</a>  <a href="chap57.html">57</a>  <a href="chap58.html">58</a>  <a href="chap59.html">59</a>  <a href="chap60.html">60</a>  <a href="chap61.html">61</a>  <a href="chap62.html">62</a>  <a href="chap63.html">63</a>  <a href="chap64.html">64</a>  <a href="chap65.html">65</a>  <a href="chap66.html">66</a>  <a href="chap67.html">67</a>  <a href="chap68.html">68</a>  <a href="chap69.html">69</a>  <a href="chap70.html">70</a>  <a href="chap71.html">71</a>  <a href="chap72.html">72</a>  <a href="chap73.html">73</a>  <a href="chap74.html">74</a>  <a href="chap75.html">75</a>  <a href="chap76.html">76</a>  <a href="chap77.html">77</a>  <a href="chap78.html">78</a>  <a href="chap79.html">79</a>  <a href="chap80.html">80</a>  <a href="chap81.html">81</a>  <a href="chap82.html">82</a>  <a href="chap83.html">83</a>  <a href="chap84.html">84</a>  <a href="chap85.html">85</a>  <a href="chap86.html">86</a>  <a href="chap87.html">87</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<div class="chlinkprevnexttop">&nbsp;<a href="chap0.html">[Top of Book]</a>&nbsp;  <a href="chap0.html#contents">[Contents]</a>&nbsp;  &nbsp;<a href="chap38.html">[Previous Chapter]</a>&nbsp;  &nbsp;<a href="chap40.html">[Next Chapter]</a>&nbsp;  </div>

<p id="mathjaxlink" class="pcenter"><a href="chap39_mj.html">[MathJax on]</a></p>
<p><a id="X8716635F7951801B" name="X8716635F7951801B"></a></p>
<div class="ChapSects"><a href="chap39.html#X8716635F7951801B">39 <span class="Heading">Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X822370B47DEA37B1">39.1 <span class="Heading">Group Elements</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X86A022F9800121F8">39.2 <span class="Heading">Creating Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7D8E473384DE9CD4">39.2-1 Group</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7F81960287F3E32A">39.2-2 GroupByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8589EF9C7B658B94">39.2-3 GroupWithGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X79C44528864044C5">39.2-4 GeneratorsOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7A0747F17B50D967">39.2-5 AsGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7E4143A08040BB47">39.2-6 ConjugateGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7939B3177BBD61E4">39.2-7 IsGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X845874BA82E1A11F">39.2-8 InfoGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7BA181CA81D785BB">39.3 <span class="Heading">Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7C82AA387A42DCA0">39.3-1 Subgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X86DC933779B1CABD">39.3-2 <span class="Heading">Index (<strong class="pkg">GAP</strong> operation)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8014135884DCC53E">39.3-3 IndexInWholeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7904AC9D7E9A3BB7">39.3-4 AsSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7839D8927E778334">39.3-5 IsSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X838186F9836F678C">39.3-6 IsNormal</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8390B5117A10CC52">39.3-7 IsCharacteristicSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X84F5464983655590">39.3-8 ConjugateSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7D9990EB837075A4">39.3-9 ConjugateSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X82ABF80780CC27AF">39.3-10 IsSubnormal</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X829766158665FB54">39.3-11 SubgroupByProperty</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7E95101F80583E77">39.3-12 SubgroupShell</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7B855B0485C3C6C5">39.4 <span class="Heading">Closures of (Sub)groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7D13FC1F8576FFD8">39.4-1 ClosureGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X81A20A397C308483">39.4-2 ClosureGroupAddElm</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X82F59F6680D1B0D5">39.4-3 ClosureGroupDefault</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7A7AF14A8052F055">39.4-4 ClosureSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7E19F92284F6684E">39.5 <span class="Heading">Expressing Group Elements as Words in Generators</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7FE8A3B08458A1BF">39.5-1 EpimorphismFromFreeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8357294D7B164106">39.5-2 Factorization</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X871508DD808EB487">39.5-3 GrowthFunctionOfGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X87BF1B887C91CA2E">39.6 <span class="Heading">Structure Descriptions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8199B74B84446971">39.6-1 StructureDescription</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X81002AA87DDBC02F">39.7 <span class="Heading">Cosets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8412ABD57986B9FC">39.7-1 RightCoset</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X835F48248571364F">39.7-2 RightCosets</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X85884F177B5D98AE">39.7-3 CanonicalRightCosetElement</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7D7625A1861D9DAB">39.7-4 IsRightCoset</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X82F6ABE378B928D1">39.7-5 CosetDecomposition</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X83C723878230D616">39.8 <span class="Heading">Transversals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X85C65D06822E716F">39.8-1 RightTransversal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X78B98B257E981046">39.9 <span class="Heading">Double Cosets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7E51ED757D17254B">39.9-1 DoubleCoset</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7F53DABD79BA4F72">39.9-2 RepresentativesContainedRightCosets</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7A5EFABB86E6D4D5">39.9-3 DoubleCosets</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X85ED464F878EF24C">39.9-4 IsDoubleCoset</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7A25B1C886CF8C6A">39.9-5 DoubleCosetRepsAndSizes</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X84AE7EE77E5FB30E">39.9-6 InfoCoset</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7D474F8F87E4E5D9">39.10 <span class="Heading">Conjugacy Classes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7B2F207F7F85F5B8">39.10-1 ConjugacyClass</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X871B570284BBA685">39.10-2 ConjugacyClasses</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7D6ED84C86C2979B">39.10-3 ConjugacyClassesByRandomSearch</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X852B3634789D770E">39.10-4 ConjugacyClassesByOrbits</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8733F87B7E4C9903">39.10-5 NrConjugacyClasses</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7BD2A4427B7FE248">39.10-6 RationalClass</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X81E9EF0A811072E8">39.10-7 RationalClasses</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X877691247DE23386">39.10-8 GaloisGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X83DD148D7DA2ABA9">39.10-9 <span class="Heading">IsConjugate</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X81A92F828400FC8A">39.10-10 NthRootsInGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X804F0F037F06E25E">39.11 <span class="Heading">Normal Structure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X87B5370C7DFD401D">39.11-1 <span class="Heading">Normalizer</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7C4E00297E37AA44">39.11-2 Core</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7CF497C77B1E8938">39.11-3 PCore</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7BDEA0A98720D1BB">39.11-4 NormalClosure</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7D25E7DC7834A703">39.11-5 NormalIntersection</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X811B8A4683DDE1F9">39.11-6 ComplementClassesRepresentatives</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8581F4E77B11C610">39.11-7 InfoComplement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7C39EE3E836D6BC6">39.12 <span class="Heading">Specific and Parametrized Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X829759F67D4247CA">39.12-1 TrivialSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7A9A3D5578CE33A0">39.12-2 CommutatorSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7CC17CF179ED7EF2">39.12-3 DerivedSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7B10B58F83DDE56E">39.12-4 CommutatorLength</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X780552B57C30DD8F">39.12-5 FittingSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X788C856C82243274">39.12-6 FrattiniSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X81D86CCE84193E4F">39.12-7 PrefrattiniSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X83D5C8B8865C85F1">39.12-8 PerfectResiduum</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X787F5F14844FAACE">39.12-9 RadicalGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X81F647FA83D8854F">39.12-10 Socle</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8440C61080CDAA14">39.12-11 SupersolvableResiduum</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X796DA805853FAC90">39.12-12 PRump</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7FF0BBDD80E8F6BF">39.13 <span class="Heading">Sylow Subgroups and Hall Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7AA351308787544C">39.13-1 SylowSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8605F3FE7A3B8E12">39.13-2 SylowComplement</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7EDBA19E828CD584">39.13-3 HallSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X832E8E6B8347B13F">39.13-4 SylowSystem</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X87A245E180D27147">39.13-5 ComplementSystem</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X82FE5DFD84F8A3C6">39.13-6 HallSystem</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X87AF37E980382499">39.14 <span class="Heading">Subgroups characterized by prime powers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7F069ACC83DB3374">39.14-1 Omega</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X83DB33747F069ACC">39.14-2 Agemo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7B75879B8085120A">39.15 <span class="Heading">Group Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7DA27D338374FD28">39.15-1 IsCyclic</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X813C952F80E775D4">39.15-2 IsElementaryAbelian</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X87D062608719F2CD">39.15-3 IsNilpotentGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7E3056237C6A5D43">39.15-4 NilpotencyClassOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8755147280C84DBB">39.15-5 IsPerfectGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X809C78D5877D31DF">39.15-6 IsSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7D7456077D3D1B86">39.15-7 IsPolycyclicGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7AADF2E88501B9FF">39.15-8 IsSupersolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X83977EB97A8E2290">39.15-9 IsMonomialGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7A6685D7819AEC32">39.15-10 IsSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X78CC9764803601E7">39.15-11 IsAlmostSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7C6AA6897C4409AC">39.15-12 <span class="Heading">IsomorphismTypeInfoFiniteSimpleGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8492B05B822AC58C">39.15-13 SimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X839CDD8C7AE39FD6">39.15-14 SimpleGroupsIterator</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X872E93F586F54FCE">39.15-15 SmallSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7EB47BF27D8CBF72">39.15-16 AllSmallNonabelianSimpleGroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X81E22D07871DF37E">39.15-17 IsFinitelyGeneratedGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8648EDA287829755">39.15-18 IsSubsetLocallyFiniteGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8089F18C810B7E3E">39.15-19 IsPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X87356BAA7E9E2142">39.15-20 PrimePGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X863434AD7DDE514B">39.15-21 PClassPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X840A4F937ABF15E1">39.15-22 RankPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X81130F9A7CFCF6BF">39.15-23 IsPSolvable</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X87415A8485FCF510">39.15-24 IsPNilpotent</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7F8264FA796B2B7D">39.16 <span class="Heading">Numerical Group Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X812827937F403300">39.16-1 AbelianInvariants</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7D44470C7DA59C1C">39.16-2 Exponent</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X843E0CCA8351FDF4">39.16-3 EulerianFunction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7AEDEDF67CFED672">39.17 <span class="Heading">Subgroup Series</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7BDD116F7833800F">39.17-1 ChiefSeries</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7AC93E977AC9ED58">39.17-2 ChiefSeriesThrough</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8724E15F81B51173">39.17-3 ChiefSeriesUnderAction</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7A0E7A8B8495B79D">39.17-4 SubnormalSeries</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X81CDCBD67BC98A5A">39.17-5 CompositionSeries</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X82C0D0217ACB2042">39.17-6 DisplayCompositionSeries</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7A879948834BD889">39.17-7 DerivedSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7A9AA1577CEC891F">39.17-8 DerivedLength</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X83F057E5791944D6">39.17-9 <span class="Heading">ElementaryAbelianSeries</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X782BD7A47D6B6503">39.17-10 InvariantElementaryAbelianSeries</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X879D55A67DB42676">39.17-11 LowerCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8428592E8773CD7B">39.17-12 UpperCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7809B7ED792669F3">39.17-13 PCentralSeries</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X82A34BD681F24A94">39.17-14 JenningsSeries</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7C08A8B77EC09CFF">39.17-15 DimensionsLoewyFactors</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X84112774812180DD">39.17-16 AscendingChain</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7C5029EE86D7FC96">39.17-17 IntermediateGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X781661FB78DC83B5">39.17-18 IntermediateSubgroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X84091B0A7E401E2B">39.18 <span class="Heading">Factor Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X80FC390C7F38A13F">39.18-1 NaturalHomomorphismByNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7E6EED0185B27C48">39.18-2 FactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7816FA867BF1B8ED">39.18-3 CommutatorFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7BB93B9778C5A0B2">39.18-4 MaximalAbelianQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7FC83E4C783572E7">39.18-5 HasAbelianFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7FAC018680B766B7">39.18-6 HasElementaryAbelianFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X822A3AB27919BC1E">39.18-7 CentralizerModulo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7D8EFB2F85AA24EE">39.19 <span class="Heading">Sets of Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7DDE67C67E871336">39.19-1 ConjugacyClassSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7C5BBF487977B8CD">39.19-2 IsConjugacyClassSubgroupsRep</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7E986BF48393113A">39.19-3 ConjugacyClassesSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8486C25380853F9B">39.19-4 ConjugacyClassesMaximalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X80399CD4870FFC4B">39.19-5 AllSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X798BF55C837DB188">39.19-6 MaximalSubgroupClassReps</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X861CD8DA790D81C2">39.19-7 MaximalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X80237A847E24E6CF">39.19-8 NormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X82ECAA427C987318">39.19-9 MaximalNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X86FDD9BA819F5644">39.19-10 MinimalNormalSubgroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7FA267497CFC0550">39.20 <span class="Heading">Subgroup Lattice</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7B104E2C86166188">39.20-1 LatticeSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X78928A3582882BFD">39.20-2 ClassElementLattice</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7E5DF287825EE7BA">39.20-3 DotFileLatticeSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X815CDA447C5DB285">39.20-4 MaximalSubgroupsLattice</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8138997C871EDF96">39.20-5 MinimalSupergroupsLattice</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7BA3484E7AE0A0E1">39.20-6 RepresentativesPerfectSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7B2233D180DF77A1">39.20-7 ConjugacyClassesPerfectSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7BFE573187B4BEF8">39.20-8 Zuppos</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X82C12E2C81963B23">39.20-9 InfoLattice</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X85E613D57F28AEFF">39.21 <span class="Heading">Specific Methods for Subgroup Lattice Computations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X86462A567DDBA6BC">39.21-1 LatticeByCyclicExtension</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X78918D83835A0EDF">39.21-2 InvariantSubgroupsElementaryAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7AD7804A803910AC">39.21-3 SubgroupsSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7F60BBB8874DFE40">39.21-4 SizeConsiderFunction</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X833C51BD7E7812C4">39.21-5 ExactSizeConsiderFunction</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7A2C774B7CFF3E07">39.21-6 InfoPcSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X79F894537D526B61">39.22 <span class="Heading">Special Generating Sets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X82FD78AF7F80A0E2">39.22-1 GeneratorsSmallest</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7A258CCF79552198">39.22-2 LargestElementGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X81D15723804771E2">39.22-3 MinimalGeneratingSet</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X814DBABC878D5232">39.22-4 SmallGeneratingSet</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7D1574457B152333">39.22-5 IndependentGeneratorsOfAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X86F835DA8264A0CE">39.22-6 IndependentGeneratorExponents</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X7CA0B6A27E0BE6B8">39.23 <span class="Heading">1-Cohomology</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X847BEC137A49BAF4">39.23-1 <span class="Heading">OneCocycles</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7E6438D5834ACCDA">39.23-2 OneCoboundaries</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X80400ABD7F40FAA0">39.23-3 OCOneCocycles</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X811E1CF07DABE924">39.23-4 ComplementClassesRepresentativesEA</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8199B1D27D487897">39.23-5 InfoCoh</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X80A4B0F282977074">39.24 <span class="Heading">Schur Covers and Multipliers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7F619DDA7DD6C43B">39.24-1 EpimorphismSchurCover</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7DD1E37987612042">39.24-2 SchurCover</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X792BC39D7CEB1D27">39.24-3 AbelianInvariantsMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X819E8AEC835F8CD1">39.24-4 Epicentre</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8739CD4686301A0E">39.24-5 NonabelianExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7E1C8CD77CDB9F71">39.24-6 EpimorphismNonabelianExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7BF8DB3D8300BB3F">39.24-7 IsCentralFactor</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7F4240CD782B6032">39.24-8 <span class="Heading">Covering groups of symmetric groups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7DDA6BC1824F78FD">39.24-9 BasicSpinRepresentationOfSymmetricGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X844CFFDE80F6AD15">39.24-10 SchurCoverOfSymmetricGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7E0F4896795E34FC">39.24-11 DoubleCoverOfAlternatingGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap39.html#X865722987E0E19B6">39.25 <span class="Heading">Tests for the Availability of Methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X798F13EA810FB215">39.25-1 CanEasilyTestMembership</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7C2A89607BDFD920">39.25-2 CanEasilyComputeWithIndependentGensAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X83245C82835D496C">39.25-3 CanComputeSize</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X8268965487364912">39.25-4 CanComputeSizeAnySubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X82DDE00D82A32083">39.25-5 CanComputeIndex</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X7BE7C36B84C23511">39.25-6 CanComputeIsSubset</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap39.html#X87D62C2C7C375E2D">39.25-7 KnowsHowToDecompose</a></span>
</div></div>
</div>

<h3>39 <span class="Heading">Groups</span></h3>

<p>This chapter explains how to create groups and defines operations for groups, that is operations whose definition does not depend on the representation used. However methods for these operations in most cases will make use of the representation.</p>

<p>If not otherwise specified, in all examples in this chapter the group <code class="code">g</code> will be the symmetric group <span class="SimpleMath">S_4</span> acting on the letters <span class="SimpleMath">{ 1, ..., 4 }</span>.</p>

<p><a id="X822370B47DEA37B1" name="X822370B47DEA37B1"></a></p>

<h4>39.1 <span class="Heading">Group Elements</span></h4>

<p>Groups in <strong class="pkg">GAP</strong> are written multiplicatively. The elements from which a group can be generated must permit multiplication and multiplicative inversion (see <a href="chap31.html#X7B97A0307EA161E5"><span class="RefLink">31.14</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=(1,2,3);;b:=(2,3,4);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">One(a);</span>
()
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Inverse(b);</span>
(2,4,3)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a*b;</span>
(1,3)(2,4)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Order(a*b);</span>
2
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Order( [ [ 1, 1 ], [ 0, 1 ] ] );</span>
infinity
</pre></div>

<p>The next example may run into an infinite loop because the given matrix in fact has infinite order.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Order( [ [ 1, 1 ], [ 0, 1 ] ] * Indeterminate( Rationals ) );</span>
#I  Order: warning, order of &lt;mat&gt; might be infinite
</pre></div>

<p>Since groups are domains, the recommended command to compute the order of a group is <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>). For convenience, group orders can also be computed with <code class="func">Order</code> (<a href="chap31.html#X84F59A2687C62763"><span class="RefLink">31.10-10</span></a>).</p>

<p>The operation <code class="func">Comm</code> (<a href="chap31.html#X80761843831B468E"><span class="RefLink">31.12-3</span></a>) can be used to compute the commutator of two elements, the operation <code class="func">LeftQuotient</code> (<a href="chap31.html#X7A37082878DB3930"><span class="RefLink">31.12-2</span></a>) computes the product <span class="SimpleMath">x^{-1} y</span>.</p>

<p><a id="X86A022F9800121F8" name="X86A022F9800121F8"></a></p>

<h4>39.2 <span class="Heading">Creating Groups</span></h4>

<p>When groups are created from generators, this means that the generators must be elements that can be multiplied and inverted (see also <a href="chap31.html#X82039A218274826F"><span class="RefLink">31.3</span></a>). For creating a free group on a set of symbols, see <code class="func">FreeGroup</code> (<a href="chap37.html#X8215999E835290F0"><span class="RefLink">37.2-1</span></a>).</p>

<p><a id="X7D8E473384DE9CD4" name="X7D8E473384DE9CD4"></a></p>

<h5>39.2-1 Group</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Group</code>( <var class="Arg">gen</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Group</code>( <var class="Arg">gens</var>[, <var class="Arg">id</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">Group( <var class="Arg">gen</var>, ... )</code> is the group generated by the arguments <var class="Arg">gen</var>, ...</p>

<p>If the only argument <var class="Arg">gens</var> is a list that is not a matrix then <code class="code">Group( <var class="Arg">gens</var> )</code> is the group generated by the elements of that list.</p>

<p>If there are two arguments, a list <var class="Arg">gens</var> and an element <var class="Arg">id</var>, then <code class="code">Group( <var class="Arg">gens</var>, <var class="Arg">id</var> )</code> is the group generated by the elements of <var class="Arg">gens</var>, with identity <var class="Arg">id</var>.</p>

<p>Note that the value of the attribute <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>) need not be equal to the list <var class="Arg">gens</var> of generators entered as argument. Use <code class="func">GroupWithGenerators</code> (<a href="chap39.html#X8589EF9C7B658B94"><span class="RefLink">39.2-3</span></a>) if you want to be sure that the argument <var class="Arg">gens</var> is stored as value of <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));</span>
Group([ (1,2,3,4), (1,2) ])
</pre></div>

<p><a id="X7F81960287F3E32A" name="X7F81960287F3E32A"></a></p>

<h5>39.2-2 GroupByGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GroupByGenerators</code>( <var class="Arg">gens</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GroupByGenerators</code>( <var class="Arg">gens</var>, <var class="Arg">id</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">GroupByGenerators</code> returns the group <span class="SimpleMath">G</span> generated by the list <var class="Arg">gens</var>. If a second argument <var class="Arg">id</var> is present then this is stored as the identity element of the group.</p>

<p>The value of the attribute <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>) of <span class="SimpleMath">G</span> need not be equal to <var class="Arg">gens</var>. <code class="func">GroupByGenerators</code> is the underlying operation called by <code class="func">Group</code> (<a href="chap39.html#X7D8E473384DE9CD4"><span class="RefLink">39.2-1</span></a>).</p>

<p><a id="X8589EF9C7B658B94" name="X8589EF9C7B658B94"></a></p>

<h5>39.2-3 GroupWithGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GroupWithGenerators</code>( <var class="Arg">gens</var>[, <var class="Arg">id</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">GroupWithGenerators</code> returns the group <span class="SimpleMath">G</span> generated by the list <var class="Arg">gens</var>. If a second argument <var class="Arg">id</var> is present then this is stored as the identity element of the group. The value of the attribute <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>) of <span class="SimpleMath">G</span> is equal to <var class="Arg">gens</var>.</p>

<p><a id="X79C44528864044C5" name="X79C44528864044C5"></a></p>

<h5>39.2-4 GeneratorsOfGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GeneratorsOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of generators of the group <var class="Arg">G</var>. If <var class="Arg">G</var> has been created by the command <code class="func">GroupWithGenerators</code> (<a href="chap39.html#X8589EF9C7B658B94"><span class="RefLink">39.2-3</span></a>) with argument <var class="Arg">gens</var>, then the list returned by <code class="func">GeneratorsOfGroup</code> will be equal to <var class="Arg">gens</var>. For such a group, each generator can also be accessed using the <code class="code">.</code> operator (see <code class="func">GeneratorsOfDomain</code> (<a href="chap31.html#X7E353DD1838AB223"><span class="RefLink">31.9-2</span></a>)): for a positive integer <span class="SimpleMath">i</span>, <code class="code"><var class="Arg">G</var>.i</code> returns the <span class="SimpleMath">i</span>-th element of the list returned by <code class="func">GeneratorsOfGroup</code>. Moreover, if <var class="Arg">G</var> is a free group, and <code class="code">name</code> is the name of a generator of <var class="Arg">G</var> then <code class="code"><var class="Arg">G</var>.name</code> also returns this generator.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=GroupWithGenerators([(1,2,3,4),(1,2)]);</span>
Group([ (1,2,3,4), (1,2) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GeneratorsOfGroup(g);</span>
[ (1,2,3,4), (1,2) ]
</pre></div>

<p>While in this example <strong class="pkg">GAP</strong> displays the group via the generating set stored in the attribute <code class="func">GeneratorsOfGroup</code>, the methods installed for <code class="func">View</code> (<a href="chap6.html#X851902C583B84CDC"><span class="RefLink">6.3-3</span></a>) will in general display only some information about the group which may even be just the fact that it is a group.</p>

<p><a id="X7A0747F17B50D967" name="X7A0747F17B50D967"></a></p>

<h5>39.2-5 AsGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AsGroup</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>if the elements of the collection <var class="Arg">D</var> form a group the command returns this group, otherwise it returns <code class="keyw">fail</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AsGroup([(1,2)]);</span>
fail
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AsGroup([(),(1,2)]);</span>
Group([ (1,2) ])
</pre></div>

<p><a id="X7E4143A08040BB47" name="X7E4143A08040BB47"></a></p>

<h5>39.2-6 ConjugateGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugateGroup</code>( <var class="Arg">G</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the conjugate group of <var class="Arg">G</var>, obtained by applying the conjugating element <var class="Arg">obj</var>.</p>

<p>To form a conjugate (group) by any object acting via <code class="code">^</code>, one can also use the infix operator <code class="code">^</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ConjugateGroup(g,(1,5));</span>
Group([ (2,3,4,5), (2,5) ])
</pre></div>

<p><a id="X7939B3177BBD61E4" name="X7939B3177BBD61E4"></a></p>

<h5>39.2-7 IsGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsGroup</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A group is a magma-with-inverses (see <code class="func">IsMagmaWithInverses</code> (<a href="chap35.html#X82CBFF648574B830"><span class="RefLink">35.1-4</span></a>)) and associative (see <code class="func">IsAssociative</code> (<a href="chap35.html#X7C83B5A47FD18FB7"><span class="RefLink">35.4-7</span></a>)) multiplication.</p>

<p><code class="code">IsGroup</code> tests whether the object <var class="Arg">obj</var> fulfills these conditions, it does <em>not</em> test whether <var class="Arg">obj</var> is a set of elements that forms a group under multiplication; use <code class="func">AsGroup</code> (<a href="chap39.html#X7A0747F17B50D967"><span class="RefLink">39.2-5</span></a>) if you want to perform such a test. (See <a href="chap13.html#X7CC6903E78F24167"><span class="RefLink">13.3</span></a> for details about categories.)</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsGroup(g);</span>
true
</pre></div>

<p><a id="X845874BA82E1A11F" name="X845874BA82E1A11F"></a></p>

<h5>39.2-8 InfoGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InfoGroup</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>is the info class for the generic group theoretic functions (see <a href="chap7.html#X7A9C902479CB6F7C"><span class="RefLink">7.4</span></a>).</p>

<p><a id="X7BA181CA81D785BB" name="X7BA181CA81D785BB"></a></p>

<h4>39.3 <span class="Heading">Subgroups</span></h4>

<p>For the general concept of parents and subdomains, see <a href="chap31.html#X7CBDD36E7B7BE286"><span class="RefLink">31.7</span></a> and <a href="chap31.html#X7B58FDEF80338DD6"><span class="RefLink">31.8</span></a>. More functions that construct certain subgroups can be found in the sections <a href="chap39.html#X804F0F037F06E25E"><span class="RefLink">39.11</span></a>, <a href="chap39.html#X7C39EE3E836D6BC6"><span class="RefLink">39.12</span></a>, <a href="chap39.html#X7FF0BBDD80E8F6BF"><span class="RefLink">39.13</span></a>, and <a href="chap39.html#X87AF37E980382499"><span class="RefLink">39.14</span></a>.</p>

<p>If a group <span class="SimpleMath">U</span> is created as a subgroup of another group <span class="SimpleMath">G</span>, <span class="SimpleMath">G</span> becomes the parent of <span class="SimpleMath">U</span>. There is no "universal" parent group, parent-child chains can be arbitrary long. <strong class="pkg">GAP</strong> stores the result of some operations (such as <code class="func">Normalizer</code> (<a href="chap39.html#X87B5370C7DFD401D"><span class="RefLink">39.11-1</span></a>)) with the parent as an attribute.</p>

<p><a id="X7C82AA387A42DCA0" name="X7C82AA387A42DCA0"></a></p>

<h5>39.3-1 Subgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Subgroup</code>( <var class="Arg">G</var>, <var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SubgroupNC</code>( <var class="Arg">G</var>, <var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Subgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates the subgroup <var class="Arg">U</var> of <var class="Arg">G</var> generated by <var class="Arg">gens</var>. The <code class="func">Parent</code> (<a href="chap31.html#X7BC856CC7F116BB0"><span class="RefLink">31.7-1</span></a>) value of <var class="Arg">U</var> will be <var class="Arg">G</var>. The <code class="code">NC</code> version does not check, whether the elements in <var class="Arg">gens</var> actually lie in <var class="Arg">G</var>.</p>

<p>The unary version of <code class="func">Subgroup</code> creates a (shell) subgroup that does not even know generators but can be used to collect information about a particular subgroup over time.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u:=Subgroup(g,[(1,2,3),(1,2)]);</span>
Group([ (1,2,3), (1,2) ])
</pre></div>

<p><a id="X86DC933779B1CABD" name="X86DC933779B1CABD"></a></p>

<h5>39.3-2 <span class="Heading">Index (<strong class="pkg">GAP</strong> operation)</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Index</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IndexNC</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a subgroup <var class="Arg">U</var> of the group <var class="Arg">G</var>, <code class="func">Index</code> returns the index <span class="SimpleMath">[<var class="Arg">G</var>:<var class="Arg">U</var>] = |<var class="Arg">G</var>| / |<var class="Arg">U</var>|</span> of <var class="Arg">U</var> in <var class="Arg">G</var>. The <code class="code">NC</code> version does not test whether <var class="Arg">U</var> is contained in <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Index(g,u);</span>
4
</pre></div>

<p><a id="X8014135884DCC53E" name="X8014135884DCC53E"></a></p>

<h5>39.3-3 IndexInWholeGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IndexInWholeGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If the family of elements of <var class="Arg">G</var> itself forms a group <var class="Arg">P</var>, this attribute returns the index of <var class="Arg">G</var> in <var class="Arg">P</var>. It is used primarily for free groups or finitely presented groups.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">freegp:=FreeGroup(1);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">freesub:=Subgroup(freegp,[freegp.1^5]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IndexInWholeGroup(freesub);</span>
5
</pre></div>

<p><a id="X7904AC9D7E9A3BB7" name="X7904AC9D7E9A3BB7"></a></p>

<h5>39.3-4 AsSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AsSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>creates a subgroup of <var class="Arg">G</var> which contains the same elements as <var class="Arg">U</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">v:=AsSubgroup(g,Group((1,2,3),(1,4)));</span>
Group([ (1,2,3), (1,4) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Parent(v);</span>
Group([ (1,2,3,4), (1,2) ])
</pre></div>

<p><a id="X7839D8927E778334" name="X7839D8927E778334"></a></p>

<h5>39.3-5 IsSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">IsSubgroup</code> returns <code class="keyw">true</code> if <var class="Arg">U</var> is a group that is a subset of the domain <var class="Arg">G</var>. This is actually checked by calling <code class="code">IsGroup( <var class="Arg">U</var> )</code> and <code class="code">IsSubset( <var class="Arg">G</var>, <var class="Arg">U</var> )</code>; note that special methods for <code class="func">IsSubset</code> (<a href="chap30.html#X79CA175481F8105F"><span class="RefLink">30.5-1</span></a>) are available that test only generators of <var class="Arg">U</var> if <var class="Arg">G</var> is closed under the group operations. So in most cases, for example whenever one knows already that <var class="Arg">U</var> is a group, it is better to call only <code class="func">IsSubset</code> (<a href="chap30.html#X79CA175481F8105F"><span class="RefLink">30.5-1</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsSubgroup(g,u);</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">v:=Group((1,2,3),(1,2));</span>
Group([ (1,2,3), (1,2) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u=v;</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsSubgroup(g,v);</span>
true
</pre></div>

<p><a id="X838186F9836F678C" name="X838186F9836F678C"></a></p>

<h5>39.3-6 IsNormal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsNormal</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the group <var class="Arg">G</var> normalizes the group <var class="Arg">U</var> and <code class="keyw">false</code> otherwise.</p>

<p>A group <var class="Arg">G</var> <em>normalizes</em> a group <var class="Arg">U</var> if and only if for every <span class="SimpleMath">g ∈ <var class="Arg">G</var></span> and <span class="SimpleMath">u ∈ <var class="Arg">U</var></span> the element <span class="SimpleMath">u^g</span> is a member of <var class="Arg">U</var>. Note that <var class="Arg">U</var> need not be a subgroup of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsNormal(g,u);</span>
false
</pre></div>

<p><a id="X8390B5117A10CC52" name="X8390B5117A10CC52"></a></p>

<h5>39.3-7 IsCharacteristicSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsCharacteristicSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>tests whether <var class="Arg">N</var> is invariant under all automorphisms of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsCharacteristicSubgroup(g,u);</span>
false
</pre></div>

<p><a id="X84F5464983655590" name="X84F5464983655590"></a></p>

<h5>39.3-8 ConjugateSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugateSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a group <var class="Arg">G</var> which has a parent group <code class="code">P</code> (see <code class="func">Parent</code> (<a href="chap31.html#X7BC856CC7F116BB0"><span class="RefLink">31.7-1</span></a>)), returns the subgroup of <code class="code">P</code>, obtained by conjugating <var class="Arg">G</var> using the conjugating element <var class="Arg">g</var>.</p>

<p>If <var class="Arg">G</var> has no parent group, it just delegates to the call to <code class="func">ConjugateGroup</code> (<a href="chap39.html#X7E4143A08040BB47"><span class="RefLink">39.2-6</span></a>) with the same arguments.</p>

<p>To form a conjugate (subgroup) by any object acting via <code class="code">^</code>, one can also use the infix operator <code class="code">^</code>.</p>

<p><a id="X7D9990EB837075A4" name="X7D9990EB837075A4"></a></p>

<h5>39.3-9 ConjugateSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugateSubgroups</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a list of all images of the group <var class="Arg">U</var> under conjugation action by <var class="Arg">G</var>.</p>

<p><a id="X82ABF80780CC27AF" name="X82ABF80780CC27AF"></a></p>

<h5>39.3-10 IsSubnormal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsSubnormal</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A subgroup <var class="Arg">U</var> of the group <var class="Arg">G</var> is subnormal if it is contained in a subnormal series of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsSubnormal(g,Group((1,2,3)));</span>
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsSubnormal(g,Group((1,2)(3,4)));</span>
true
</pre></div>

<p><a id="X829766158665FB54" name="X829766158665FB54"></a></p>

<h5>39.3-11 SubgroupByProperty</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SubgroupByProperty</code>( <var class="Arg">G</var>, <var class="Arg">prop</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates a subgroup of <var class="Arg">G</var> consisting of those elements fulfilling <var class="Arg">prop</var> (which is a tester function). No test is done whether the property actually defines a subgroup.</p>

<p>Note that currently very little functionality beyond an element test exists for groups created this way.</p>

<p><a id="X7E95101F80583E77" name="X7E95101F80583E77"></a></p>

<h5>39.3-12 SubgroupShell</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SubgroupShell</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates a subgroup of <var class="Arg">G</var> which at this point is not yet specified further (but will be later, for example by assigning a generating set).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u:=SubgroupByProperty(g,i-&gt;3^i=3);</span>
&lt;subgrp of Group([ (1,2,3,4), (1,2) ]) by property&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">(1,3) in u; (1,4) in u; (1,5) in u;</span>
false
true
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GeneratorsOfGroup(u);</span>
[ (1,2), (1,4,2) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u:=SubgroupShell(g);</span>
&lt;group&gt;
</pre></div>

<p><a id="X7B855B0485C3C6C5" name="X7B855B0485C3C6C5"></a></p>

<h4>39.4 <span class="Heading">Closures of (Sub)groups</span></h4>

<p><a id="X7D13FC1F8576FFD8" name="X7D13FC1F8576FFD8"></a></p>

<h5>39.4-1 ClosureGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ClosureGroup</code>( <var class="Arg">G</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>creates the group generated by the elements of <var class="Arg">G</var> and <var class="Arg">obj</var>. <var class="Arg">obj</var> can be either an element or a collection of elements, in particular another group.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=SmallGroup(24,12);;u:=Subgroup(g,[g.3,g.4]);</span>
Group([ f3, f4 ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ClosureGroup(u,g.2);</span>
Group([ f2, f3, f4 ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ClosureGroup(u,[g.1,g.2]);</span>
Group([ f1, f2, f3, f4 ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ClosureGroup(u,Group(g.2*g.1));</span>
Group([ f1*f2^2, f3, f4 ])
</pre></div>

<p><a id="X81A20A397C308483" name="X81A20A397C308483"></a></p>

<h5>39.4-2 ClosureGroupAddElm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ClosureGroupAddElm</code>( <var class="Arg">G</var>, <var class="Arg">elm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ClosureGroupCompare</code>( <var class="Arg">G</var>, <var class="Arg">elm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ClosureGroupIntest</code>( <var class="Arg">G</var>, <var class="Arg">elm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>These three functions together with <code class="func">ClosureGroupDefault</code> (<a href="chap39.html#X82F59F6680D1B0D5"><span class="RefLink">39.4-3</span></a>) implement the main methods for <code class="func">ClosureGroup</code> (<a href="chap39.html#X7D13FC1F8576FFD8"><span class="RefLink">39.4-1</span></a>). In the ordering given, they just add <var class="Arg">elm</var> to the generators, remove duplicates and identity elements, and test whether <var class="Arg">elm</var> is already contained in <var class="Arg">G</var>.</p>

<p><a id="X82F59F6680D1B0D5" name="X82F59F6680D1B0D5"></a></p>

<h5>39.4-3 ClosureGroupDefault</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ClosureGroupDefault</code>( <var class="Arg">G</var>, <var class="Arg">elm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This functions returns the closure of the group <var class="Arg">G</var> with the element <var class="Arg">elm</var>. If <var class="Arg">G</var> has the attribute <code class="func">AsSSortedList</code> (<a href="chap30.html#X856D927378C33548"><span class="RefLink">30.3-10</span></a>) then also the result has this attribute. This is used to implement the default method for <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) and <code class="func">EnumeratorSorted</code> (<a href="chap30.html#X80CD7DDC7D0C60D5"><span class="RefLink">30.3-3</span></a>).</p>

<p><a id="X7A7AF14A8052F055" name="X7A7AF14A8052F055"></a></p>

<h5>39.4-4 ClosureSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ClosureSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">obj</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ClosureSubgroupNC</code>( <var class="Arg">G</var>, <var class="Arg">obj</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a group <var class="Arg">G</var> that stores a parent group (see <a href="chap31.html#X7CBDD36E7B7BE286"><span class="RefLink">31.7</span></a>), <code class="func">ClosureSubgroup</code> calls <code class="func">ClosureGroup</code> (<a href="chap39.html#X7D13FC1F8576FFD8"><span class="RefLink">39.4-1</span></a>) with the same arguments; if the result is a subgroup of the parent of <var class="Arg">G</var> then the parent of <var class="Arg">G</var> is set as parent of the result, otherwise an error is raised. The check whether the result is contained in the parent of <var class="Arg">G</var> is omitted by the <code class="code">NC</code> version. As a wrong parent might imply wrong properties this version should be used with care.</p>

<p><a id="X7E19F92284F6684E" name="X7E19F92284F6684E"></a></p>

<h4>39.5 <span class="Heading">Expressing Group Elements as Words in Generators</span></h4>

<p>Using homomorphisms (see chapter <a href="chap40.html#X83702FC27B3C3098"><span class="RefLink">40</span></a>) is is possible to express group elements as words in given generators: Create a free group (see <code class="func">FreeGroup</code> (<a href="chap37.html#X8215999E835290F0"><span class="RefLink">37.2-1</span></a>)) on the correct number of generators and create a homomorphism from this free group onto the group <var class="Arg">G</var> in whose generators you want to factorize. Then the preimage of an element of <var class="Arg">G</var> is a word in the free generators, that will map on this element again.</p>

<p><a id="X7FE8A3B08458A1BF" name="X7FE8A3B08458A1BF"></a></p>

<h5>39.5-1 EpimorphismFromFreeGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; EpimorphismFromFreeGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a group <var class="Arg">G</var> with a known generating set, this attribute returns a homomorphism from a free group that maps the free generators to the groups generators.</p>

<p>The option <code class="code">names</code> can be used to prescribe a (print) name for the free generators.</p>

<p>The following example shows how to decompose elements of <span class="SimpleMath">S_4</span> in the generators <code class="code">(1,2,3,4)</code> and <code class="code">(1,2)</code>:</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));</span>
Group([ (1,2,3,4), (1,2) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">hom:=EpimorphismFromFreeGroup(g:names:=["x","y"]);</span>
[ x, y ] -&gt; [ (1,2,3,4), (1,2) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">PreImagesRepresentative(hom,(1,4));</span>
y^-1*x^-1*(x^-1*y^-1)^2*x
</pre></div>

<p>The following example stems from a real request to the <strong class="pkg">GAP</strong> Forum. In September 2000 a <strong class="pkg">GAP</strong> user working with puzzles wanted to express the permutation <code class="code">(1,2)</code> as a word as short as possible in particular generators of the symmetric group <span class="SimpleMath">S_16</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">perms := [ (1,2,3,7,11,10,9,5), (2,3,4,8,12,11,10,6),</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">  (5,6,7,11,15,14,13,9), (6,7,8,12,16,15,14,10) ];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">puzzle := Group( perms );;Size( puzzle );</span>
20922789888000
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">hom:=EpimorphismFromFreeGroup(puzzle:names:=["a", "b", "c", "d"]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">word := PreImagesRepresentative( hom, (1,2) );</span>
a^-1*c*b*c^-1*a*b^-1*a^-2*c^-1*a*b^-1*c*b
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Length( word );</span>
13
</pre></div>

<p><a id="X8357294D7B164106" name="X8357294D7B164106"></a></p>

<h5>39.5-2 Factorization</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Factorization</code>( <var class="Arg">G</var>, <var class="Arg">elm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a factorization of <var class="Arg">elm</var> as word in the generators of the group <var class="Arg">G</var> given in the attribute <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>). The attribute <code class="func">EpimorphismFromFreeGroup</code> (<a href="chap39.html#X7FE8A3B08458A1BF"><span class="RefLink">39.5-1</span></a>) of <var class="Arg">G</var> will contain a map from the group <var class="Arg">G</var> to the free group in which the word is expressed. The attribute <code class="func">MappingGeneratorsImages</code> (<a href="chap40.html#X863805187A24B5E3"><span class="RefLink">40.10-2</span></a>) of this map gives a list of generators and corresponding letters.</p>

<p>The algorithm used forms all elements of the group to ensure a short word is found. Therefore this function should <em>not</em> be used when the group <var class="Arg">G</var> has more than a few million elements. Because of this, one should not call this function within algorithms, but use homomorphisms instead.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G:=SymmetricGroup( 6 );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">r:=(3,4);; s:=(1,2,3,4,5,6);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput"># create subgroup to force the system to use the generators r and s:</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">H:= Subgroup(G, [ r, s ] );</span>
Group([ (3,4), (1,2,3,4,5,6) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Factorization( H, (1,2,3) );</span>
(x2*x1)^2*x2^-2
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">s*r*s*r*s^-2;</span>
(1,2,3)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">MappingGeneratorsImages(EpimorphismFromFreeGroup(H));</span>
[ [ x1, x2 ], [ (3,4), (1,2,3,4,5,6) ] ]
</pre></div>

<p><a id="X871508DD808EB487" name="X871508DD808EB487"></a></p>

<h5>39.5-3 GrowthFunctionOfGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GrowthFunctionOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GrowthFunctionOfGroup</code>( <var class="Arg">G</var>, <var class="Arg">radius</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a group <var class="Arg">G</var> with a generating set given in <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>), this function calculates the number of elements whose shortest expression as words in the generating set is of a particular length. It returns a list <var class="Arg">L</var>, whose <span class="SimpleMath">i+1</span> entry counts the number of elements whose shortest word expression has length <span class="SimpleMath">i</span>. If a maximal length <var class="Arg">radius</var> is given, only words up to length <var class="Arg">radius</var> are counted. Otherwise the group must be finite and all elements are enumerated.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GrowthFunctionOfGroup(MathieuGroup(12));  </span>
[ 1, 5, 19, 70, 255, 903, 3134, 9870, 25511, 38532, 16358, 382 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GrowthFunctionOfGroup(MathieuGroup(12),2);</span>
[ 1, 5, 19 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GrowthFunctionOfGroup(MathieuGroup(12),99);</span>
[ 1, 5, 19, 70, 255, 903, 3134, 9870, 25511, 38532, 16358, 382 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">free:=FreeGroup("a","b");</span>
&lt;free group on the generators [ a, b ]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">product:=free/ParseRelators(free,"a2,b3");</span>
&lt;fp group on the generators [ a, b ]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetIsFinite(product,false);</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GrowthFunctionOfGroup(product,10);</span>
[ 1, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64 ]
</pre></div>

<p><a id="X87BF1B887C91CA2E" name="X87BF1B887C91CA2E"></a></p>

<h4>39.6 <span class="Heading">Structure Descriptions</span></h4>

<p><a id="X8199B74B84446971" name="X8199B74B84446971"></a></p>

<h5>39.6-1 StructureDescription</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; StructureDescription</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The method for <code class="func">StructureDescription</code> exhibits a structure of the given group <var class="Arg">G</var> to some extent, using the strategy outlined below. The idea is to return a possibly short string which gives some insight in the structure of the considered group. It is intended primarily for small groups (order less than 100) or groups with few normal subgroups, in other cases, in particular large <span class="SimpleMath">p</span>-groups, it can be very costly. Furthermore, the string returned is -- as the action on chief factors is not described -- often not the most useful way to describe a group.</p>

<p>The string returned by <code class="func">StructureDescription</code> is <strong class="button">not</strong> an isomorphism invariant: non-isomorphic groups can have the same string value, and two isomorphic groups in different representations can produce different strings. The value returned by <code class="func">StructureDescription</code> is a string of the following form:</p>


<div class="example"><pre>
  StructureDescription(&lt;G&gt;) ::=
      1                                 ; trivial group 
    | C&lt;size&gt;                           ; cyclic group
    | A&lt;degree&gt;                         ; alternating group
    | S&lt;degree&gt;                         ; symmetric group
    | D&lt;size&gt;                           ; dihedral group
    | Q&lt;size&gt;                           ; quaternion group
    | QD&lt;size&gt;                          ; quasidihedral group
    | PSL(&lt;n&gt;,&lt;q&gt;)                      ; projective special linear group
    | SL(&lt;n&gt;,&lt;q&gt;)                       ; special linear group
    | GL(&lt;n&gt;,&lt;q&gt;)                       ; general linear group
    | PSU(&lt;n&gt;,&lt;q&gt;)                      ; proj. special unitary group
    | O(2&lt;n&gt;+1,&lt;q&gt;)                     ; orthogonal group, type B
    | O+(2&lt;n&gt;,&lt;q&gt;)                      ; orthogonal group, type D
    | O-(2&lt;n&gt;,&lt;q&gt;)                      ; orthogonal group, type 2D
    | PSp(2&lt;n&gt;,&lt;q&gt;)                     ; proj. special symplectic group
    | Sz(&lt;q&gt;)                           ; Suzuki group
    | Ree(&lt;q&gt;)                          ; Ree group (type 2F or 2G)
    | E(6,&lt;q&gt;) | E(7,&lt;q&gt;) | E(8,&lt;q&gt;)    ; Lie group of exceptional type
    | 2E(6,&lt;q&gt;) | F(4,&lt;q&gt;) | G(2,&lt;q&gt;)
    | 3D(4,&lt;q&gt;)                         ; Steinberg triality group
    | M11 | M12 | M22 | M23 | M24
    | J1 | J2 | J3 | J4 | Co1 | Co2
    | Co3 | Fi22 | Fi23 | Fi24' | Suz
    | HS | McL | He | HN | Th | B
    | M | ON | Ly | Ru                  ; sporadic simple group
    | 2F(4,2)'                          ; Tits group
    | PerfectGroup(&lt;size&gt;,&lt;id&gt;)         ; the indicated group from the
                                        ; library of perfect groups
    | A x B                             ; direct product
    | N : H                             ; semidirect product
    | C(G) . G/C(G) = G' . G/G'         ; non-split extension
                                        ; (equal alternatives and
                                        ; trivial extensions omitted)
    | Phi(G) . G/Phi(G)                 ; non-split extension:
                                        ; Frattini subgroup and
                                        ; Frattini factor group
</pre></div>

<p>Note that the <code class="func">StructureDescription</code> is only <em>one</em> possible way of building up the given group from smaller pieces.</p>

<p>The option "short" is recognized - if this option is set, an abbreviated output format is used (e.g. <code class="code">"6x3"</code> instead of <code class="code">"C6 x C3"</code>).</p>

<p>If the <code class="func">Name</code> (<a href="chap12.html#X7F14EF9D81432113"><span class="RefLink">12.8-2</span></a>) attribute is not bound, but <code class="func">StructureDescription</code> is, <code class="func">View</code> (<a href="chap6.html#X851902C583B84CDC"><span class="RefLink">6.3-3</span></a>) prints the value of the attribute <code class="func">StructureDescription</code>. The <code class="func">Print</code> (<a href="chap6.html#X7AFA64D97A1F39A3"><span class="RefLink">6.3-4</span></a>)ed representation of a group is not affected by computing a <code class="func">StructureDescription</code>.</p>

<p>The strategy used to compute a <code class="func">StructureDescription</code> is as follows:</p>


<dl>
<dt><strong class="Mark">1.</strong></dt>
<dd><p>Lookup in a precomputed list, if the order of <var class="Arg">G</var> is not larger than 100 and not equal to 64.</p>

</dd>
<dt><strong class="Mark">2.</strong></dt>
<dd><p>If <var class="Arg">G</var> is abelian, then decompose it into cyclic factors in "elementary divisors style". For example, <code class="code">"C2 x C3 x C3"</code> is <code class="code">"C6 x C3"</code>.</p>

</dd>
<dt><strong class="Mark">3.</strong></dt>
<dd><p>Recognize alternating groups, symmetric groups, dihedral groups, quasidihedral groups, quaternion groups, PSL's, SL's, GL's and simple groups not listed so far as basic building blocks.</p>

</dd>
<dt><strong class="Mark">4.</strong></dt>
<dd><p>Decompose <var class="Arg">G</var> into a direct product of irreducible factors.</p>

</dd>
<dt><strong class="Mark">5.</strong></dt>
<dd><p>Recognize semidirect products <var class="Arg">G</var>=<span class="SimpleMath">N</span>:<span class="SimpleMath">H</span>, where <span class="SimpleMath">N</span> is normal. Select a pair <span class="SimpleMath">N</span>, <span class="SimpleMath">H</span> with the following preferences:</p>


<dl>
<dt><strong class="Mark">1.</strong></dt>
<dd><p><span class="SimpleMath">H</span> is abelian</p>

</dd>
<dt><strong class="Mark">2.</strong></dt>
<dd><p><span class="SimpleMath">N</span> is abelian</p>

</dd>
<dt><strong class="Mark">2a.</strong></dt>
<dd><p><span class="SimpleMath">N</span> has many abelian invariants</p>

</dd>
<dt><strong class="Mark">3.</strong></dt>
<dd><p><span class="SimpleMath">N</span> is a direct product</p>

</dd>
<dt><strong class="Mark">3a.</strong></dt>
<dd><p><span class="SimpleMath">N</span> has many direct factors</p>

</dd>
<dt><strong class="Mark">4.</strong></dt>
<dd><p><span class="SimpleMath">ϕ: H →</span> Aut(<span class="SimpleMath">N</span>), <span class="SimpleMath">h ↦ (n ↦ n^h)</span> is injective.</p>

</dd>
</dl>
</dd>
<dt><strong class="Mark">6.</strong></dt>
<dd><p>Fall back to non-splitting extensions: If the centre or the commutator factor group is non-trivial, write <var class="Arg">G</var> as Z(<var class="Arg">G</var>).<var class="Arg">G</var>/Z(<var class="Arg">G</var>) or <var class="Arg">G</var>'.<var class="Arg">G</var>/<var class="Arg">G</var>', respectively. Otherwise if the Frattini subgroup is non-trivial, write <var class="Arg">G</var> as <span class="SimpleMath">Φ</span>(<var class="Arg">G</var>).<var class="Arg">G</var>/<span class="SimpleMath">Φ</span>(<var class="Arg">G</var>).</p>

</dd>
<dt><strong class="Mark">7.</strong></dt>
<dd><p>If no decomposition is found (maybe this is not the case for any finite group), try to identify <var class="Arg">G</var> in the perfect groups library. If this fails also, then return a string describing this situation.</p>

</dd>
</dl>
<p>Note that <code class="func">StructureDescription</code> is <em>not</em> intended to be a research tool, but rather an educational tool. The reasons for this are as follows:</p>


<dl>
<dt><strong class="Mark">1.</strong></dt>
<dd><p>"Most" groups do not have "nice" decompositions. This is in some contrast to what is often taught in elementary courses on group theory, where it is sometimes suggested that basically every group can be written as iterated direct or semidirect product of cyclic groups and nonabelian simple groups.</p>

</dd>
<dt><strong class="Mark">2.</strong></dt>
<dd><p>In particular many <span class="SimpleMath">p</span>-groups have very "similar" structure, and <code class="func">StructureDescription</code> can only exhibit a little of it. Changing this would likely make the output not essentially easier to read than a pc presentation.</p>

</dd>
</dl>

<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">l := AllSmallGroups(12);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(l,StructureDescription);; l;</span>
[ C3 : C4, C12, A4, D12, C6 x C2 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(AllSmallGroups(40),G-&gt;StructureDescription(G:short));</span>
[ "5:8", "40", "5:8", "5:Q8", "4xD10", "D40", "2x(5:4)", "(10x2):2", 
  "20x2", "5xD8", "5xQ8", "2x(5:4)", "2^2xD10", "10x2^2" ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(AllTransitiveGroups(DegreeAction,6),</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">        G-&gt;StructureDescription(G:short));</span>
[ "6", "S3", "D12", "A4", "3xS3", "2xA4", "S4", "S4", "S3xS3", 
  "(3^2):4", "2xS4", "A5", "(S3xS3):2", "S5", "A6", "S6" ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">StructureDescription(PSL(4,2));</span>
"A8"
</pre></div>

<p><a id="X81002AA87DDBC02F" name="X81002AA87DDBC02F"></a></p>

<h4>39.7 <span class="Heading">Cosets</span></h4>

<p><a id="X8412ABD57986B9FC" name="X8412ABD57986B9FC"></a></p>

<h5>39.7-1 RightCoset</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RightCoset</code>( <var class="Arg">U</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the right coset of <var class="Arg">U</var> with representative <var class="Arg">g</var>, which is the set of all elements of the form <span class="SimpleMath">ug</span> for all <span class="SimpleMath">u ∈ <var class="Arg">U</var></span>. <var class="Arg">g</var> must be an element of a larger group <var class="Arg">G</var> which contains <var class="Arg">U</var>. For element operations such as <code class="keyw">in</code> a right coset behaves like a set of group elements.</p>

<p>Right cosets are external orbits for the action of <var class="Arg">U</var> which acts via <code class="func">OnLeftInverse</code> (<a href="chap41.html#X832DF5327ECA0E44"><span class="RefLink">41.2-3</span></a>). Of course the action of a larger group <var class="Arg">G</var> on right cosets is via <code class="func">OnRight</code> (<a href="chap41.html#X7960924D84B5B18F"><span class="RefLink">41.2-2</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u:=Group((1,2,3), (1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">c:=RightCoset(u,(2,3,4));</span>
RightCoset(Group( [ (1,2,3), (1,2) ] ),(2,3,4))
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ActingDomain(c);</span>
Group([ (1,2,3), (1,2) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Representative(c);</span>
(2,3,4)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(c);</span>
6
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AsList(c);</span>
[ (2,3,4), (1,4,2), (1,3,4,2), (1,3)(2,4), (2,4), (1,4,2,3) ]
</pre></div>

<p><a id="X835F48248571364F" name="X835F48248571364F"></a></p>

<h5>39.7-2 RightCosets</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RightCosets</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RightCosetsNC</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes a duplicate free list of right cosets <var class="Arg">U</var> <span class="SimpleMath">g</span> for <span class="SimpleMath">g ∈</span> <var class="Arg">G</var>. A set of representatives for the elements in this list forms a right transversal of <var class="Arg">U</var> in <var class="Arg">G</var>. (By inverting the representatives one obtains a list of representatives of the left cosets of <var class="Arg">U</var>.) The <code class="code">NC</code> version does not check whether <var class="Arg">U</var> is a subgroup of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RightCosets(g,u);</span>
[ RightCoset(Group( [ (1,2,3), (1,2) ] ),()), 
  RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,3)(2,4)), 
  RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,4)(2,3)), 
  RightCoset(Group( [ (1,2,3), (1,2) ] ),(1,2)(3,4)) ]
</pre></div>

<p><a id="X85884F177B5D98AE" name="X85884F177B5D98AE"></a></p>

<h5>39.7-3 CanonicalRightCosetElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanonicalRightCosetElement</code>( <var class="Arg">U</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a "canonical" representative of the right coset <var class="Arg">U</var> <var class="Arg">g</var> which is independent of the given representative <var class="Arg">g</var>. This can be used to compare cosets by comparing their canonical representatives.</p>

<p>The representative chosen to be the "canonical" one is representation dependent and only guaranteed to remain the same within one <strong class="pkg">GAP</strong> session.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CanonicalRightCosetElement(u,(2,4,3));</span>
(3,4)
</pre></div>

<p><a id="X7D7625A1861D9DAB" name="X7D7625A1861D9DAB"></a></p>

<h5>39.7-4 IsRightCoset</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsRightCoset</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>The category of right cosets.</p>

<p><strong class="pkg">GAP</strong> does not provide left cosets as a separate data type, but as the left coset <span class="SimpleMath">gU</span> consists of exactly the inverses of the elements of the right coset <span class="SimpleMath">Ug^{-1}</span> calculations with left cosets can be emulated using right cosets by inverting the representatives.</p>

<p><a id="X82F6ABE378B928D1" name="X82F6ABE378B928D1"></a></p>

<h5>39.7-5 CosetDecomposition</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CosetDecomposition</code>( <var class="Arg">G</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a finite group <var class="Arg">G</var> and a subgroup <span class="SimpleMath"><var class="Arg">S</var>le<var class="Arg">G</var></span> this function returns a partition of the elements of <var class="Arg">G</var> according to the (right) cosets of <var class="Arg">S</var>. The result is a list of lists, each sublist corresponding to one coset. The first sublist is the elements list of the subgroup, the other lists are arranged accordingly.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CosetDecomposition(SymmetricGroup(4),SymmetricGroup(3));          </span>
[ [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ], 
  [ (1,4), (1,4)(2,3), (1,2,4), (1,2,3,4), (1,3,2,4), (1,3,4) ], 
  [ (1,4,2), (1,4,2,3), (2,4), (2,3,4), (1,3)(2,4), (1,3,4,2) ], 
  [ (1,4,3), (1,4,3,2), (1,2,4,3), (1,2)(3,4), (2,4,3), (3,4) ] ]
</pre></div>

<p><a id="X83C723878230D616" name="X83C723878230D616"></a></p>

<h4>39.8 <span class="Heading">Transversals</span></h4>

<p><a id="X85C65D06822E716F" name="X85C65D06822E716F"></a></p>

<h5>39.8-1 RightTransversal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RightTransversal</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A right transversal <span class="SimpleMath">t</span> is a list of representatives for the set <span class="SimpleMath"><var class="Arg">U</var><var class="Arg">G</var></span> of right cosets (consisting of cosets <span class="SimpleMath">Ug</span>) of <span class="SimpleMath">U</span> in <span class="SimpleMath">G</span>.</p>

<p>The object returned by <code class="func">RightTransversal</code> is not a plain list, but an object that behaves like an immutable list of length <span class="SimpleMath">[<var class="Arg">G</var>:<var class="Arg">U</var>]</span>, except if <var class="Arg">U</var> is the trivial subgroup of <var class="Arg">G</var> in which case <code class="func">RightTransversal</code> may return the sorted plain list of coset representatives.</p>

<p>The operation <code class="func">PositionCanonical</code> (<a href="chap21.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>), called for a transversal <span class="SimpleMath">t</span> and an element <span class="SimpleMath">g</span> of <var class="Arg">G</var>, will return the position of the representative in <span class="SimpleMath">t</span> that lies in the same coset of <var class="Arg">U</var> as the element <span class="SimpleMath">g</span> does. (In comparison, <code class="func">Position</code> (<a href="chap21.html#X79975EC6783B4293"><span class="RefLink">21.16-1</span></a>) will return <code class="keyw">fail</code> if the element is not equal to the representative.) Functions that implement group actions such as <code class="func">Action</code> (<a href="chap41.html#X85A8E93D786C3C9C"><span class="RefLink">41.7-2</span></a>) or <code class="func">Permutation</code> (<a href="chap41.html#X7807A33381DCAB26"><span class="RefLink">41.9-1</span></a>) (see Chapter <a href="chap41.html#X87115591851FB7F4"><span class="RefLink">41</span></a>) use <code class="func">PositionCanonical</code> (<a href="chap21.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>), therefore it is possible to "act" on a right transversal to implement the action on the cosets. This is often much more efficient than acting on cosets.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u:=Subgroup(g,[(1,2,3),(1,2)]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">rt:=RightTransversal(g,u);</span>
RightTransversal(Group([ (1,2,3,4), (1,2) ]),Group([ (1,2,3), (1,2) ]))
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Length(rt);</span>
4
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Position(rt,(1,2,3));</span>
fail
</pre></div>

<p>Note that the elements of a right transversal are not necessarily "canonical" in the sense of <code class="func">CanonicalRightCosetElement</code> (<a href="chap39.html#X85884F177B5D98AE"><span class="RefLink">39.7-3</span></a>), but we may compute a list of canonical coset representatives by calling that function. (See also <code class="func">PositionCanonical</code> (<a href="chap21.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>).)</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(RightTransversal(g,u),i-&gt;CanonicalRightCosetElement(u,i));</span>
[ (), (2,3,4), (1,2,3,4), (3,4) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">PositionCanonical(rt,(1,2,3));</span>
1
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">rt[1];</span>
()
</pre></div>

<p><a id="X78B98B257E981046" name="X78B98B257E981046"></a></p>

<h4>39.9 <span class="Heading">Double Cosets</span></h4>

<p><a id="X7E51ED757D17254B" name="X7E51ED757D17254B"></a></p>

<h5>39.9-1 DoubleCoset</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DoubleCoset</code>( <var class="Arg">U</var>, <var class="Arg">g</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The groups <var class="Arg">U</var> and <var class="Arg">V</var> must be subgroups of a common supergroup <var class="Arg">G</var> of which <var class="Arg">g</var> is an element. This command constructs the double coset <var class="Arg">U</var> <var class="Arg">g</var> <var class="Arg">V</var> which is the set of all elements of the form <span class="SimpleMath">ugv</span> for any <span class="SimpleMath">u ∈ <var class="Arg">U</var></span>, <span class="SimpleMath">v ∈ <var class="Arg">V</var></span>. For element operations such as <code class="keyw">in</code>, a double coset behaves like a set of group elements. The double coset stores <var class="Arg">U</var> in the attribute <code class="code">LeftActingGroup</code>, <var class="Arg">g</var> as <code class="func">Representative</code> (<a href="chap30.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>), and <var class="Arg">V</var> as <code class="code">RightActingGroup</code>.</p>

<p><a id="X7F53DABD79BA4F72" name="X7F53DABD79BA4F72"></a></p>

<h5>39.9-2 RepresentativesContainedRightCosets</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RepresentativesContainedRightCosets</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A double coset <span class="SimpleMath"><var class="Arg">D</var> = U g V</span> can be considered as a union of right cosets <span class="SimpleMath">U h_i</span>. (It is the union of the orbit of <span class="SimpleMath">U g</span> under right multiplication by <span class="SimpleMath">V</span>.) For a double coset <var class="Arg">D</var> this function returns a set of representatives <span class="SimpleMath">h_i</span> such that <var class="Arg">D</var> <span class="SimpleMath">= ⋃_{h_i} U h_i</span>. The representatives returned are canonical for <span class="SimpleMath">U</span> (see <code class="func">CanonicalRightCosetElement</code> (<a href="chap39.html#X85884F177B5D98AE"><span class="RefLink">39.7-3</span></a>)) and form a set.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u:=Subgroup(g,[(1,2,3),(1,2)]);;v:=Subgroup(g,[(3,4)]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">c:=DoubleCoset(u,(2,4),v);</span>
DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(2,4),Group( [ (3,4) ] ))
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">(1,2,3) in c;</span>
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">(2,3,4) in c;</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftActingGroup(c);</span>
Group([ (1,2,3), (1,2) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RightActingGroup(c);</span>
Group([ (3,4) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RepresentativesContainedRightCosets(c);</span>
[ (2,3,4) ]
</pre></div>

<p><a id="X7A5EFABB86E6D4D5" name="X7A5EFABB86E6D4D5"></a></p>

<h5>39.9-3 DoubleCosets</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DoubleCosets</code>( <var class="Arg">G</var>, <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DoubleCosetsNC</code>( <var class="Arg">G</var>, <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes a duplicate free list of all double cosets <var class="Arg">U</var> <span class="SimpleMath">g</span> <var class="Arg">V</var> for <span class="SimpleMath">g ∈ <var class="Arg">G</var></span>. The groups <var class="Arg">U</var> and <var class="Arg">V</var> must be subgroups of the group <var class="Arg">G</var>. The <code class="code">NC</code> version does not check whether <var class="Arg">U</var> and <var class="Arg">V</var> are subgroups of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">dc:=DoubleCosets(g,u,v);</span>
[ DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(),Group( [ (3,4) ] )), 
  DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(1,3)(2,4),Group( 
    [ (3,4) ] )), DoubleCoset(Group( [ (1,2,3), (1,2) ] ),(1,4)
    (2,3),Group( [ (3,4) ] )) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(dc,Representative);</span>
[ (), (1,3)(2,4), (1,4)(2,3) ]
</pre></div>

<p><a id="X85ED464F878EF24C" name="X85ED464F878EF24C"></a></p>

<h5>39.9-4 IsDoubleCoset</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsDoubleCoset</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>The category of double cosets.</p>

<p><a id="X7A25B1C886CF8C6A" name="X7A25B1C886CF8C6A"></a></p>

<h5>39.9-5 DoubleCosetRepsAndSizes</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DoubleCosetRepsAndSizes</code>( <var class="Arg">G</var>, <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a list of double coset representatives and their sizes, the entries are lists of the form <span class="SimpleMath">[ r, n ]</span> where <span class="SimpleMath">r</span> and <span class="SimpleMath">n</span> are an element of the double coset and the size of the coset, respectively. This operation is faster than <code class="func">DoubleCosetsNC</code> (<a href="chap39.html#X7A5EFABB86E6D4D5"><span class="RefLink">39.9-3</span></a>) because no double coset objects have to be created.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">dc:=DoubleCosetRepsAndSizes(g,u,v);</span>
[ [ (), 12 ], [ (1,3)(2,4), 6 ], [ (1,4)(2,3), 6 ] ]
</pre></div>

<p><a id="X84AE7EE77E5FB30E" name="X84AE7EE77E5FB30E"></a></p>

<h5>39.9-6 InfoCoset</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InfoCoset</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>The information function for coset and double coset operations is <code class="func">InfoCoset</code>.</p>

<p><a id="X7D474F8F87E4E5D9" name="X7D474F8F87E4E5D9"></a></p>

<h4>39.10 <span class="Heading">Conjugacy Classes</span></h4>

<p><a id="X7B2F207F7F85F5B8" name="X7B2F207F7F85F5B8"></a></p>

<h5>39.10-1 ConjugacyClass</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugacyClass</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>creates the conjugacy class in <var class="Arg">G</var> with representative <var class="Arg">g</var>. This class is an external set, so functions such as <code class="func">Representative</code> (<a href="chap30.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>) (which returns <var class="Arg">g</var>), <code class="func">ActingDomain</code> (<a href="chap41.html#X7B9DB15D80CE28B4"><span class="RefLink">41.12-3</span></a>) (which returns <var class="Arg">G</var>), <code class="func">StabilizerOfExternalSet</code> (<a href="chap41.html#X7BAFF02B7D6DF9F2"><span class="RefLink">41.12-10</span></a>) (which returns the centralizer of <var class="Arg">g</var>) and <code class="func">AsList</code> (<a href="chap30.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) work for it.</p>

<p>A conjugacy class is an external orbit (see <code class="func">ExternalOrbit</code> (<a href="chap41.html#X7FB656AE7A066C35"><span class="RefLink">41.12-9</span></a>)) of group elements with the group acting by conjugation on it. Thus element tests or operation representatives can be computed. The attribute <code class="func">Centralizer</code> (<a href="chap35.html#X7DE33AFC823C7873"><span class="RefLink">35.4-4</span></a>) gives the centralizer of the representative (which is the same result as <code class="func">StabilizerOfExternalSet</code> (<a href="chap41.html#X7BAFF02B7D6DF9F2"><span class="RefLink">41.12-10</span></a>)). (This is a slight abuse of notation: This is <em>not</em> the centralizer of the class as a <em>set</em> which would be the standard behaviour of <code class="func">Centralizer</code> (<a href="chap35.html#X7DE33AFC823C7873"><span class="RefLink">35.4-4</span></a>).)</p>

<p><a id="X871B570284BBA685" name="X871B570284BBA685"></a></p>

<h5>39.10-2 ConjugacyClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugacyClasses</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the conjugacy classes of elements of <var class="Arg">G</var> as a list of class objects of <var class="Arg">G</var> (see <code class="func">ConjugacyClass</code> (<a href="chap39.html#X7B2F207F7F85F5B8"><span class="RefLink">39.10-1</span></a>) for details). It is guaranteed that the class of the identity is in the first position, the further arrangement depends on the method chosen (and might be different for equal but not identical groups).</p>

<p>For very small groups (of size up to 500) the classes will be computed by the conjugation action of <var class="Arg">G</var> on itself (see <code class="func">ConjugacyClassesByOrbits</code> (<a href="chap39.html#X852B3634789D770E"><span class="RefLink">39.10-4</span></a>)). This can be deliberately switched off using the "<code class="code">noaction</code>" option shown below.</p>

<p>For solvable groups, the default method to compute the classes is by homomorphic lift (see section <a href="chap45.html#X79DCCF6D80351859"><span class="RefLink">45.17</span></a>).</p>

<p>For other groups the method of <a href="chapBib.html#biBHulpkeClasses">[Hul00]</a> is employed.</p>

<p><code class="func">ConjugacyClasses</code> supports the following options that can be used to modify this strategy:</p>


<dl>
<dt><strong class="Mark"><code class="code">random</code></strong></dt>
<dd><p>The classes are computed by random search. See <code class="func">ConjugacyClassesByRandomSearch</code> (<a href="chap39.html#X7D6ED84C86C2979B"><span class="RefLink">39.10-3</span></a>) below.</p>

</dd>
<dt><strong class="Mark"><code class="code">action</code></strong></dt>
<dd><p>The classes are computed by action of <var class="Arg">G</var> on itself. See <code class="func">ConjugacyClassesByOrbits</code> (<a href="chap39.html#X852B3634789D770E"><span class="RefLink">39.10-4</span></a>) below.</p>

</dd>
<dt><strong class="Mark"><code class="code">noaction</code></strong></dt>
<dd><p>Even for small groups <code class="func">ConjugacyClassesByOrbits</code> (<a href="chap39.html#X852B3634789D770E"><span class="RefLink">39.10-4</span></a>) is not used as a default. This can be useful if the elements of the group use a lot of memory.</p>

</dd>
</dl>

<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">cl:=ConjugacyClasses(g);</span>
[ ()^G, (1,2)^G, (1,2)(3,4)^G, (1,2,3)^G, (1,2,3,4)^G ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Representative(cl[3]);Centralizer(cl[3]);</span>
(1,2)(3,4)
Group([ (1,2), (1,3)(2,4), (3,4) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Centralizer(cl[5]));</span>
4
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(cl[2]);</span>
6
</pre></div>

<p>In general, you will not need to have to influence the method, but simply call <code class="func">ConjugacyClasses</code><strong class="pkg">GAP</strong> will try to select a suitable method on its own. The method specifications are provided here mainly for expert use.</p>

<p><a id="X7D6ED84C86C2979B" name="X7D6ED84C86C2979B"></a></p>

<h5>39.10-3 ConjugacyClassesByRandomSearch</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugacyClassesByRandomSearch</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the classes of the group <var class="Arg">G</var> by random search. This works very efficiently for almost simple groups.</p>

<p>This function is also accessible via the option <code class="code">random</code> to the function <code class="func">ConjugacyClass</code> (<a href="chap39.html#X7B2F207F7F85F5B8"><span class="RefLink">39.10-1</span></a>).</p>

<p><a id="X852B3634789D770E" name="X852B3634789D770E"></a></p>

<h5>39.10-4 ConjugacyClassesByOrbits</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugacyClassesByOrbits</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the classes of the group <var class="Arg">G</var> as orbits of <var class="Arg">G</var> on its elements. This can be quick but unsurprisingly may also take a lot of memory if <var class="Arg">G</var> becomes larger. All the classes will store their element list and thus a membership test will be quick as well.</p>

<p>This function is also accessible via the option <code class="code">action</code> to the function <code class="func">ConjugacyClass</code> (<a href="chap39.html#X7B2F207F7F85F5B8"><span class="RefLink">39.10-1</span></a>).</p>

<p>Typically, for small groups (roughly of order up to <span class="SimpleMath">10^3</span>) the computation of classes as orbits under the action is fastest; memory restrictions (and the increasing cost of eliminating duplicates) make this less efficient for larger groups.</p>

<p>Calculation by random search has the smallest memory requirement, but in generally performs worse, the more classes are there.</p>

<p>The following example shows the effect of this for a small group with many classes:</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h:=Group((4,5)(6,7,8),(1,2,3)(5,6,9));;ConjugacyClasses(h:noaction);;time;</span>
110
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h:=Group((4,5)(6,7,8),(1,2,3)(5,6,9));;ConjugacyClasses(h:random);;time;</span>
300
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h:=Group((4,5)(6,7,8),(1,2,3)(5,6,9));;ConjugacyClasses(h:action);;time;</span>
30
</pre></div>

<p><a id="X8733F87B7E4C9903" name="X8733F87B7E4C9903"></a></p>

<h5>39.10-5 NrConjugacyClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NrConjugacyClasses</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the number of conjugacy classes of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NrConjugacyClasses(g);</span>
5
</pre></div>

<p><a id="X7BD2A4427B7FE248" name="X7BD2A4427B7FE248"></a></p>

<h5>39.10-6 RationalClass</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RationalClass</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>creates the rational class in <var class="Arg">G</var> with representative <var class="Arg">g</var>. A rational class consists of all elements that are conjugate to <var class="Arg">g</var> or to an <span class="SimpleMath">i</span>-th power of <var class="Arg">g</var> where <span class="SimpleMath">i</span> is coprime to the order of <span class="SimpleMath">g</span>. Thus a rational class can be interpreted as a conjugacy class of cyclic subgroups. A rational class is an external set (<code class="func">IsExternalSet</code> (<a href="chap41.html#X8264C3C479FF0A8B"><span class="RefLink">41.12-1</span></a>)) of group elements with the group acting by conjugation on it, but not an external orbit.</p>

<p><a id="X81E9EF0A811072E8" name="X81E9EF0A811072E8"></a></p>

<h5>39.10-7 RationalClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RationalClasses</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of the rational classes of the group <var class="Arg">G</var>. (See <code class="func">RationalClass</code> (<a href="chap39.html#X7BD2A4427B7FE248"><span class="RefLink">39.10-6</span></a>).)</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RationalClasses(DerivedSubgroup(g));</span>
[ RationalClass( AlternatingGroup( [ 1 .. 4 ] ), () ), 
  RationalClass( AlternatingGroup( [ 1 .. 4 ] ), (1,2)(3,4) ), 
  RationalClass( AlternatingGroup( [ 1 .. 4 ] ), (1,2,3) ) ]
</pre></div>

<p><a id="X877691247DE23386" name="X877691247DE23386"></a></p>

<h5>39.10-8 GaloisGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GaloisGroup</code>( <var class="Arg">ratcl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Suppose that <var class="Arg">ratcl</var> is a rational class of a group <span class="SimpleMath">G</span> with representative <span class="SimpleMath">g</span>. The exponents <span class="SimpleMath">i</span> for which <span class="SimpleMath">g^i</span> lies already in the ordinary conjugacy class of <span class="SimpleMath">g</span>, form a subgroup of the <em>prime residue class group</em> <span class="SimpleMath">P_n</span> (see <code class="func">PrimitiveRootMod</code> (<a href="chap15.html#X82440BB9812FF148"><span class="RefLink">15.3-3</span></a>)), the so-called <em>Galois group</em> of the rational class. The prime residue class group <span class="SimpleMath">P_n</span> is obtained in <strong class="pkg">GAP</strong> as <code class="code">Units( Integers mod <var class="Arg">n</var> )</code>, the unit group of a residue class ring. The Galois group of a rational class <var class="Arg">ratcl</var> is stored in the attribute <code class="func">GaloisGroup</code> as a subgroup of this group.</p>

<p><a id="X83DD148D7DA2ABA9" name="X83DD148D7DA2ABA9"></a></p>

<h5>39.10-9 <span class="Heading">IsConjugate</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsConjugate</code>( <var class="Arg">G</var>, <var class="Arg">x</var>, <var class="Arg">y</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsConjugate</code>( <var class="Arg">G</var>, <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>tests whether the elements <var class="Arg">x</var> and <var class="Arg">y</var> or the subgroups <var class="Arg">U</var> and <var class="Arg">V</var> are conjugate under the action of <var class="Arg">G</var>. (They do not need to be <em>contained in</em> <var class="Arg">G</var>.) This command is only a shortcut to <code class="func">RepresentativeAction</code> (<a href="chap41.html#X857DC7B085EB0539"><span class="RefLink">41.6-1</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsConjugate(g,Group((1,2,3,4),(1,3)),Group((1,3,2,4),(1,2)));</span>
true
</pre></div>

<p><code class="func">RepresentativeAction</code> (<a href="chap41.html#X857DC7B085EB0539"><span class="RefLink">41.6-1</span></a>) can be used to obtain conjugating elements.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RepresentativeAction(g,(1,2),(3,4));</span>
(1,3)(2,4)
</pre></div>

<p><a id="X81A92F828400FC8A" name="X81A92F828400FC8A"></a></p>

<h5>39.10-10 NthRootsInGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NthRootsInGroup</code>( <var class="Arg">G</var>, <var class="Arg">e</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">e</var> be an element in the group <var class="Arg">G</var>. This function returns a list of all those elements in <var class="Arg">G</var> whose <var class="Arg">n</var>-th power is <var class="Arg">e</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NthRootsInGroup(g,(1,2)(3,4),2);</span>
[ (1,3,2,4), (1,4,2,3) ]
</pre></div>

<p><a id="X804F0F037F06E25E" name="X804F0F037F06E25E"></a></p>

<h4>39.11 <span class="Heading">Normal Structure</span></h4>

<p>For the operations <code class="func">Centralizer</code> (<a href="chap35.html#X7DE33AFC823C7873"><span class="RefLink">35.4-4</span></a>) and <code class="func">Centre</code> (<a href="chap35.html#X847ABE6F781C7FE8"><span class="RefLink">35.4-5</span></a>), see Chapter <a href="chap35.html#X873E502F7D21C39C"><span class="RefLink">35</span></a>.</p>

<p><a id="X87B5370C7DFD401D" name="X87B5370C7DFD401D"></a></p>

<h5>39.11-1 <span class="Heading">Normalizer</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Normalizer</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Normalizer</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For two groups <var class="Arg">G</var>, <var class="Arg">U</var>, <code class="func">Normalizer</code> computes the normalizer <span class="SimpleMath">N_<var class="Arg">G</var>(<var class="Arg">U</var>)</span>, that is, the stabilizer of <var class="Arg">U</var> under the conjugation action of <var class="Arg">G</var>.</p>

<p>For a group <var class="Arg">G</var> and a group element <var class="Arg">g</var>, <code class="func">Normalizer</code> computes <span class="SimpleMath">N_<var class="Arg">G</var>(⟨ <var class="Arg">g</var> ⟩)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Normalizer(g,Subgroup(g,[(1,2,3)]));</span>
Group([ (1,2,3), (2,3) ])
</pre></div>

<p><a id="X7C4E00297E37AA44" name="X7C4E00297E37AA44"></a></p>

<h5>39.11-2 Core</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Core</code>( <var class="Arg">S</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">S</var> and <var class="Arg">U</var> are groups of elements in the same family, this operation returns the core of <var class="Arg">U</var> in <var class="Arg">S</var>, that is the intersection of all <var class="Arg">S</var>-conjugates of <var class="Arg">U</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Core(g,Subgroup(g,[(1,2,3,4)]));</span>
Group(())
</pre></div>

<p><a id="X7CF497C77B1E8938" name="X7CF497C77B1E8938"></a></p>

<h5>39.11-3 PCore</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PCore</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The <em><var class="Arg">p</var>-core</em> of <var class="Arg">G</var> is the largest normal <var class="Arg">p</var>-subgroup of <var class="Arg">G</var>. It is the core of a Sylow <var class="Arg">p</var> subgroup of <var class="Arg">G</var>, see <code class="func">Core</code> (<a href="chap39.html#X7C4E00297E37AA44"><span class="RefLink">39.11-2</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">PCore(g,2);</span>
Group([ (1,4)(2,3), (1,2)(3,4) ])
</pre></div>

<p><a id="X7BDEA0A98720D1BB" name="X7BDEA0A98720D1BB"></a></p>

<h5>39.11-4 NormalClosure</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NormalClosure</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The normal closure of <var class="Arg">U</var> in <var class="Arg">G</var> is the smallest normal subgroup of the closure of <var class="Arg">G</var> and <var class="Arg">U</var> which contains <var class="Arg">U</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NormalClosure(g,Subgroup(g,[(1,2,3)]));</span>
Group([ (1,2,3), (2,3,4) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NormalClosure(g,Group((3,4,5)));</span>
Group([ (3,4,5), (1,5,4), (1,2,5) ])
</pre></div>

<p><a id="X7D25E7DC7834A703" name="X7D25E7DC7834A703"></a></p>

<h5>39.11-5 NormalIntersection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NormalIntersection</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes the intersection of <var class="Arg">G</var> and <var class="Arg">U</var>, assuming that <var class="Arg">G</var> is normalized by <var class="Arg">U</var>. This works faster than <code class="code">Intersection</code>, but will not produce the intersection if <var class="Arg">G</var> is not normalized by <var class="Arg">U</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NormalIntersection(Group((1,2)(3,4),(1,3)(2,4)),Group((1,2,3,4)));</span>
Group([ (1,3)(2,4) ])
</pre></div>

<p><a id="X811B8A4683DDE1F9" name="X811B8A4683DDE1F9"></a></p>

<h5>39.11-6 ComplementClassesRepresentatives</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ComplementClassesRepresentatives</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">N</var> be a normal subgroup of <var class="Arg">G</var>. This command returns a set of representatives for the conjugacy classes of complements of <var class="Arg">N</var> in <var class="Arg">G</var>. Complements are subgroups of <var class="Arg">G</var> which intersect trivially with <var class="Arg">N</var> and together with <var class="Arg">N</var> generate <var class="Arg">G</var>.</p>

<p>At the moment only methods for a solvable <var class="Arg">N</var> are available.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ComplementClassesRepresentatives(g,Group((1,2)(3,4),(1,3)(2,4)));</span>
[ Group([ (3,4), (2,4,3) ]) ]
</pre></div>

<p><a id="X8581F4E77B11C610" name="X8581F4E77B11C610"></a></p>

<h5>39.11-7 InfoComplement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InfoComplement</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>Info class for the complement routines.</p>

<p><a id="X7C39EE3E836D6BC6" name="X7C39EE3E836D6BC6"></a></p>

<h4>39.12 <span class="Heading">Specific and Parametrized Subgroups</span></h4>

<p>The centre of a group (the subgroup of those elements that commute with all other elements of the group) can be computed by the operation <code class="func">Centre</code> (<a href="chap35.html#X847ABE6F781C7FE8"><span class="RefLink">35.4-5</span></a>).</p>

<p><a id="X829759F67D4247CA" name="X829759F67D4247CA"></a></p>

<h5>39.12-1 TrivialSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; TrivialSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>

<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">TrivialSubgroup(g);</span>
Group(())
</pre></div>

<p><a id="X7A9A3D5578CE33A0" name="X7A9A3D5578CE33A0"></a></p>

<h5>39.12-2 CommutatorSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CommutatorSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">G</var> and <var class="Arg">H</var> are two groups of elements in the same family, this operation returns the group generated by all commutators <span class="SimpleMath">[ g, h ] = g^{-1} h^{-1} g h</span> (see <code class="func">Comm</code> (<a href="chap31.html#X80761843831B468E"><span class="RefLink">31.12-3</span></a>)) of elements <span class="SimpleMath">g ∈ <var class="Arg">G</var></span> and <span class="SimpleMath">h ∈ <var class="Arg">H</var></span>, that is the group <span class="SimpleMath">⟨ [ g, h ] ∣ g ∈ <var class="Arg">G</var>, h ∈ <var class="Arg">H</var></span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CommutatorSubgroup(Group((1,2,3),(1,2)),Group((2,3,4),(3,4)));</span>
Group([ (1,4)(2,3), (1,3,4) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(last);</span>
12
</pre></div>

<p><a id="X7CC17CF179ED7EF2" name="X7CC17CF179ED7EF2"></a></p>

<h5>39.12-3 DerivedSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DerivedSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The derived subgroup <span class="SimpleMath"><var class="Arg">G</var>'</span> of <var class="Arg">G</var> is the subgroup generated by all commutators of pairs of elements of <var class="Arg">G</var>. It is normal in <var class="Arg">G</var> and the factor group <span class="SimpleMath"><var class="Arg">G</var>/<var class="Arg">G</var>'</span> is the largest abelian factor group of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">DerivedSubgroup(g);</span>
Group([ (1,3,2), (2,4,3) ])
</pre></div>

<p><a id="X7B10B58F83DDE56E" name="X7B10B58F83DDE56E"></a></p>

<h5>39.12-4 CommutatorLength</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CommutatorLength</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the minimal number <span class="SimpleMath">n</span> such that each element in the derived subgroup (see <code class="func">DerivedSubgroup</code> (<a href="chap39.html#X7CC17CF179ED7EF2"><span class="RefLink">39.12-3</span></a>)) of the group <var class="Arg">G</var> can be written as a product of (at most) <span class="SimpleMath">n</span> commutators of elements in <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CommutatorLength( g );</span>
1
</pre></div>

<p><a id="X780552B57C30DD8F" name="X780552B57C30DD8F"></a></p>

<h5>39.12-5 FittingSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FittingSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The Fitting subgroup of a group <var class="Arg">G</var> is its largest nilpotent normal subgroup.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">FittingSubgroup(g);</span>
Group([ (1,2)(3,4), (1,4)(2,3) ])
</pre></div>

<p><a id="X788C856C82243274" name="X788C856C82243274"></a></p>

<h5>39.12-6 FrattiniSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FrattiniSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The Frattini subgroup of a group <var class="Arg">G</var> is the intersection of all maximal subgroups of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">FrattiniSubgroup(g);</span>
Group(())
</pre></div>

<p><a id="X81D86CCE84193E4F" name="X81D86CCE84193E4F"></a></p>

<h5>39.12-7 PrefrattiniSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PrefrattiniSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a Prefrattini subgroup of the finite solvable group <var class="Arg">G</var>.</p>

<p>A factor <span class="SimpleMath">M/N</span> of <var class="Arg">G</var> is called a Frattini factor if <span class="SimpleMath">M/N</span> is contained in the Frattini subgroup of <span class="SimpleMath"><var class="Arg">G</var>/N</span>. A subgroup <span class="SimpleMath">P</span> is a Prefrattini subgroup of <var class="Arg">G</var> if <span class="SimpleMath">P</span> covers each Frattini chief factor of <var class="Arg">G</var>, and if for each maximal subgroup of <var class="Arg">G</var> there exists a conjugate maximal subgroup, which contains <span class="SimpleMath">P</span>. In a finite solvable group <var class="Arg">G</var> the Prefrattini subgroups form a characteristic conjugacy class of subgroups and the intersection of all these subgroups is the Frattini subgroup of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G := SmallGroup( 60, 7 );</span>
&lt;pc group of size 60 with 4 generators&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">P := PrefrattiniSubgroup(G);</span>
Group([ f2 ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(P);</span>
2
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsNilpotent(P);</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Core(G,P);</span>
Group([  ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">FrattiniSubgroup(G);</span>
Group([  ])
</pre></div>

<p><a id="X83D5C8B8865C85F1" name="X83D5C8B8865C85F1"></a></p>

<h5>39.12-8 PerfectResiduum</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PerfectResiduum</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the smallest normal subgroup of <var class="Arg">G</var> that has a solvable factor group.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">PerfectResiduum(Group((1,2,3,4,5),(1,2)));</span>
Group([ (1,3,2), (1,4,3), (1,5,4) ])
</pre></div>

<p><a id="X787F5F14844FAACE" name="X787F5F14844FAACE"></a></p>

<h5>39.12-9 RadicalGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RadicalGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the radical of <var class="Arg">G</var>, i.e., the largest solvable normal subgroup of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RadicalGroup(SL(2,5));</span>
&lt;group of 2x2 matrices of size 2 over GF(5)&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(last);</span>
2
</pre></div>

<p><a id="X81F647FA83D8854F" name="X81F647FA83D8854F"></a></p>

<h5>39.12-10 Socle</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Socle</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The socle of the group <var class="Arg">G</var> is the subgroup generated by all minimal normal subgroups.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Socle(g);</span>
Group([ (1,4)(2,3), (1,2)(3,4) ])
</pre></div>

<p><a id="X8440C61080CDAA14" name="X8440C61080CDAA14"></a></p>

<h5>39.12-11 SupersolvableResiduum</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SupersolvableResiduum</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the supersolvable residuum of the group <var class="Arg">G</var>, that is, its smallest normal subgroup <span class="SimpleMath">N</span> such that the factor group <span class="SimpleMath"><var class="Arg">G</var> / N</span> is supersolvable.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SupersolvableResiduum(g);</span>
Group([ (1,2)(3,4), (1,4)(2,3) ])
</pre></div>

<p><a id="X796DA805853FAC90" name="X796DA805853FAC90"></a></p>

<h5>39.12-12 PRump</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PRump</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a prime <span class="SimpleMath">p</span>, the <em><var class="Arg">p</var>-rump</em> of a group <var class="Arg">G</var> is the subgroup <span class="SimpleMath"><var class="Arg">G</var>' <var class="Arg">G</var>^<var class="Arg">p</var></span>.</p>

<p><em>@example missing!@</em></p>

<p><a id="X7FF0BBDD80E8F6BF" name="X7FF0BBDD80E8F6BF"></a></p>

<h4>39.13 <span class="Heading">Sylow Subgroups and Hall Subgroups</span></h4>

<p>With respect to the following <strong class="pkg">GAP</strong> functions, please note that by theorems of P. Hall, a group <span class="SimpleMath">G</span> is solvable if and only if one of the following conditions holds.</p>

<ol>
<li><p>For each prime <span class="SimpleMath">p</span> dividing the order of <span class="SimpleMath">G</span>, there exists a <span class="SimpleMath">p</span>-complement (see <code class="func">SylowComplement</code> (<a href="chap39.html#X8605F3FE7A3B8E12"><span class="RefLink">39.13-2</span></a>)).</p>

</li>
<li><p>For each set <span class="SimpleMath">P</span> of primes dividing the order of <span class="SimpleMath">G</span>, there exists a <span class="SimpleMath">P</span>-Hall subgroup (see <code class="func">HallSubgroup</code> (<a href="chap39.html#X7EDBA19E828CD584"><span class="RefLink">39.13-3</span></a>)).</p>

</li>
<li><p><span class="SimpleMath">G</span> has a Sylow system (see <code class="func">SylowSystem</code> (<a href="chap39.html#X832E8E6B8347B13F"><span class="RefLink">39.13-4</span></a>)).</p>

</li>
<li><p><span class="SimpleMath">G</span> has a complement system (see <code class="func">ComplementSystem</code> (<a href="chap39.html#X87A245E180D27147"><span class="RefLink">39.13-5</span></a>)).</p>

</li>
</ol>
<p><a id="X7AA351308787544C" name="X7AA351308787544C"></a></p>

<h5>39.13-1 SylowSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SylowSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a Sylow <var class="Arg">p</var> subgroup of the finite group <var class="Arg">G</var>. This is a <var class="Arg">p</var>-subgroup of <var class="Arg">G</var> whose index in <var class="Arg">G</var> is coprime to <var class="Arg">p</var>. <code class="func">SylowSubgroup</code> computes Sylow subgroups via the operation <code class="code">SylowSubgroupOp</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SylowSubgroup(g,2);</span>
Group([ (1,2), (3,4), (1,3)(2,4) ])
</pre></div>

<p><a id="X8605F3FE7A3B8E12" name="X8605F3FE7A3B8E12"></a></p>

<h5>39.13-2 SylowComplement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SylowComplement</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a Sylow <var class="Arg">p</var>-complement of the finite group <var class="Arg">G</var>. This is a subgroup <span class="SimpleMath">U</span> of order coprime to <var class="Arg">p</var> such that the index <span class="SimpleMath">[<var class="Arg">G</var>:U]</span> is a <var class="Arg">p</var>-power.</p>

<p>At the moment methods exist only if <var class="Arg">G</var> is solvable and <strong class="pkg">GAP</strong> will issue an error if <var class="Arg">G</var> is not solvable.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SylowComplement(g,3);</span>
Group([ (1,2), (3,4), (1,3)(2,4) ])
</pre></div>

<p><a id="X7EDBA19E828CD584" name="X7EDBA19E828CD584"></a></p>

<h5>39.13-3 HallSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; HallSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">P</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes a <var class="Arg">P</var>-Hall subgroup for a set <var class="Arg">P</var> of primes. This is a subgroup the order of which is only divisible by primes in <var class="Arg">P</var> and whose index is coprime to all primes in <var class="Arg">P</var>. Such a subgroup is unique up to conjugacy if <var class="Arg">G</var> is solvable. The function computes Hall subgroups via the operation <code class="code">HallSubgroupOp</code>.</p>

<p>If <var class="Arg">G</var> is solvable this function always returns a subgroup. If <var class="Arg">G</var> is not solvable this function might return a subgroup (if it is unique up to conjugacy), a list of subgroups (which are representatives of the conjugacy classes in case there are several such classes) or <code class="keyw">fail</code> if no such subgroup exists.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h:=SmallGroup(60,10);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u:=HallSubgroup(h,[2,3]);</span>
Group([ f1, f2, f3 ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(u);</span>
12
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h:=PSL(3,5);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HallSubgroup(h,[2,3]);  </span>
[ &lt;permutation group of size 96 with 6 generators&gt;, 
  &lt;permutation group of size 96 with 6 generators&gt; ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u := HallSubgroup(h,[3,31]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(u); StructureDescription(u);</span>
93
"C31 : C3"
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HallSubgroup(h,[5,31]);</span>
fail
</pre></div>

<p><a id="X832E8E6B8347B13F" name="X832E8E6B8347B13F"></a></p>

<h5>39.13-4 SylowSystem</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SylowSystem</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A Sylow system of a group <var class="Arg">G</var> is a set of Sylow subgroups of <var class="Arg">G</var> such that every pair of subgroups from this set commutes as subgroups. Sylow systems exist only for solvable groups. The operation returns <code class="keyw">fail</code> if the group <var class="Arg">G</var> is not solvable.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h:=SmallGroup(60,10);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SylowSystem(h);</span>
[ Group([ f1, f2 ]), Group([ f3 ]), Group([ f4 ]) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(last,Size);</span>
[ 4, 3, 5 ]
</pre></div>

<p><a id="X87A245E180D27147" name="X87A245E180D27147"></a></p>

<h5>39.13-5 ComplementSystem</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ComplementSystem</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A complement system of a group <var class="Arg">G</var> is a set of Hall <span class="SimpleMath">p'</span>-subgroups of <var class="Arg">G</var>, where <span class="SimpleMath">p'</span> runs through the subsets of prime factors of <span class="SimpleMath">|<var class="Arg">G</var>|</span> that omit exactly one prime. Every pair of subgroups from this set commutes as subgroups. Complement systems exist only for solvable groups, therefore <code class="func">ComplementSystem</code> returns <code class="keyw">fail</code> if the group <var class="Arg">G</var> is not solvable.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ComplementSystem(h);</span>
[ Group([ f3, f4 ]), Group([ f1, f2, f4 ]), Group([ f1, f2, f3 ]) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(last,Size);</span>
[ 15, 20, 12 ]
</pre></div>

<p><a id="X82FE5DFD84F8A3C6" name="X82FE5DFD84F8A3C6"></a></p>

<h5>39.13-6 HallSystem</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; HallSystem</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list containing one Hall <span class="SimpleMath">P</span>-subgroup for each set <span class="SimpleMath">P</span> of prime divisors of the order of <var class="Arg">G</var>. Hall systems exist only for solvable groups. The operation returns <code class="keyw">fail</code> if the group <var class="Arg">G</var> is not solvable.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HallSystem(h);</span>
[ Group([  ]), Group([ f1, f2 ]), Group([ f1, f2, f3 ]), 
  Group([ f1, f2, f3, f4 ]), Group([ f1, f2, f4 ]), Group([ f3 ]), 
  Group([ f3, f4 ]), Group([ f4 ]) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(last,Size);</span>
[ 1, 4, 12, 60, 20, 3, 15, 5 ]
</pre></div>

<p><a id="X87AF37E980382499" name="X87AF37E980382499"></a></p>

<h4>39.14 <span class="Heading">Subgroups characterized by prime powers</span></h4>

<p><a id="X7F069ACC83DB3374" name="X7F069ACC83DB3374"></a></p>

<h5>39.14-1 Omega</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Omega</code>( <var class="Arg">G</var>, <var class="Arg">p</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a <var class="Arg">p</var>-group <var class="Arg">G</var>, one defines <span class="SimpleMath">Ω_<var class="Arg">n</var>(<var class="Arg">G</var>) = { g ∈ <var class="Arg">G</var> ∣ g^{<var class="Arg">p</var>^<var class="Arg">n</var>} = 1 }</span>. The default value for <var class="Arg">n</var> is <code class="code">1</code>.</p>

<p><em>@At the moment methods exist only for abelian <var class="Arg">G</var> and <var class="Arg">n</var>=1.@</em></p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h:=SmallGroup(16,10);</span>
&lt;pc group of size 16 with 4 generators&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Omega(h,2);</span>
Group([ f2, f3, f4 ])
</pre></div>

<p><a id="X83DB33747F069ACC" name="X83DB33747F069ACC"></a></p>

<h5>39.14-2 Agemo</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Agemo</code>( <var class="Arg">G</var>, <var class="Arg">p</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a <var class="Arg">p</var>-group <var class="Arg">G</var>, one defines <span class="SimpleMath">℧_<var class="Arg">n</var>(G) = ⟨ g^{<var class="Arg">p</var>^<var class="Arg">n</var>} ∣ g ∈ <var class="Arg">G</var></span>. The default value for <var class="Arg">n</var> is <code class="code">1</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Agemo(h,2);Agemo(h,2,2);</span>
Group([ f4 ])
Group([  ])
</pre></div>

<p><a id="X7B75879B8085120A" name="X7B75879B8085120A"></a></p>

<h4>39.15 <span class="Heading">Group Properties</span></h4>

<p>Some properties of groups can be defined not only for groups but also for other structures. For example, nilpotency and solvability make sense also for algebras. Note that these names refer to different definitions for groups and algebras, contrary to the situation with finiteness or commutativity. In such cases, the name of the function for groups got a suffix <code class="code">Group</code> to distinguish different meanings for different structures.</p>

<p>Some functions, such as <code class="func">IsPSolvable</code> (<a href="chap39.html#X81130F9A7CFCF6BF"><span class="RefLink">39.15-23</span></a>) and <code class="func">IsPNilpotent</code> (<a href="chap39.html#X87415A8485FCF510"><span class="RefLink">39.15-24</span></a>), although they are mathematical properties, are not properties in the sense of <strong class="pkg">GAP</strong> (see <a href="chap13.html#X7C701DBF7BAE649A"><span class="RefLink">13.5</span></a> and <a href="chap13.html#X871597447BB998A1"><span class="RefLink">13.7</span></a>), as they depend on a parameter.</p>

<p><a id="X7DA27D338374FD28" name="X7DA27D338374FD28"></a></p>

<h5>39.15-1 IsCyclic</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsCyclic</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is <em>cyclic</em> if it can be generated by one element. For a cyclic group, one can compute a generating set consisting of only one element using <code class="func">MinimalGeneratingSet</code> (<a href="chap39.html#X81D15723804771E2"><span class="RefLink">39.22-3</span></a>).</p>

<p><a id="X813C952F80E775D4" name="X813C952F80E775D4"></a></p>

<h5>39.15-2 IsElementaryAbelian</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsElementaryAbelian</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group <var class="Arg">G</var> is elementary abelian if it is commutative and if there is a prime <span class="SimpleMath">p</span> such that the order of each element in <var class="Arg">G</var> divides <span class="SimpleMath">p</span>.</p>

<p><a id="X87D062608719F2CD" name="X87D062608719F2CD"></a></p>

<h5>39.15-3 IsNilpotentGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsNilpotentGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is <em>nilpotent</em> if the lower central series (see <code class="func">LowerCentralSeriesOfGroup</code> (<a href="chap39.html#X879D55A67DB42676"><span class="RefLink">39.17-11</span></a>) for a definition) reaches the trivial subgroup in a finite number of steps.</p>

<p><a id="X7E3056237C6A5D43" name="X7E3056237C6A5D43"></a></p>

<h5>39.15-4 NilpotencyClassOfGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NilpotencyClassOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The nilpotency class of a nilpotent group <var class="Arg">G</var> is the number of steps in the lower central series of <var class="Arg">G</var> (see <code class="func">LowerCentralSeriesOfGroup</code> (<a href="chap39.html#X879D55A67DB42676"><span class="RefLink">39.17-11</span></a>));</p>

<p>If <var class="Arg">G</var> is not nilpotent an error is issued.</p>

<p><a id="X8755147280C84DBB" name="X8755147280C84DBB"></a></p>

<h5>39.15-5 IsPerfectGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsPerfectGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is <em>perfect</em> if it equals its derived subgroup (see <code class="func">DerivedSubgroup</code> (<a href="chap39.html#X7CC17CF179ED7EF2"><span class="RefLink">39.12-3</span></a>)).</p>

<p><a id="X809C78D5877D31DF" name="X809C78D5877D31DF"></a></p>

<h5>39.15-6 IsSolvableGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsSolvableGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is <em>solvable</em> if the derived series (see <code class="func">DerivedSeriesOfGroup</code> (<a href="chap39.html#X7A879948834BD889"><span class="RefLink">39.17-7</span></a>) for a definition) reaches the trivial subgroup in a finite number of steps.</p>

<p>For finite groups this is the same as being polycyclic (see <code class="func">IsPolycyclicGroup</code> (<a href="chap39.html#X7D7456077D3D1B86"><span class="RefLink">39.15-7</span></a>)), and each polycyclic group is solvable, but there are infinite solvable groups that are not polycyclic.</p>

<p><a id="X7D7456077D3D1B86" name="X7D7456077D3D1B86"></a></p>

<h5>39.15-7 IsPolycyclicGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsPolycyclicGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is polycyclic if it has a subnormal series with cyclic factors. For finite groups this is the same as if the group is solvable (see <code class="func">IsSolvableGroup</code> (<a href="chap39.html#X809C78D5877D31DF"><span class="RefLink">39.15-6</span></a>)).</p>

<p><a id="X7AADF2E88501B9FF" name="X7AADF2E88501B9FF"></a></p>

<h5>39.15-8 IsSupersolvableGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsSupersolvableGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A finite group is <em>supersolvable</em> if it has a normal series with cyclic factors.</p>

<p><a id="X83977EB97A8E2290" name="X83977EB97A8E2290"></a></p>

<h5>39.15-9 IsMonomialGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsMonomialGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A finite group is <em>monomial</em> if every irreducible complex character is induced from a linear character of a subgroup.</p>

<p><a id="X7A6685D7819AEC32" name="X7A6685D7819AEC32"></a></p>

<h5>39.15-10 IsSimpleGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsSimpleGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is <em>simple</em> if it is nontrivial and has no nontrivial normal subgroups.</p>

<p><a id="X78CC9764803601E7" name="X78CC9764803601E7"></a></p>

<h5>39.15-11 IsAlmostSimpleGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsAlmostSimpleGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group <var class="Arg">G</var> is <em>almost simple</em> if a nonabelian simple group <span class="SimpleMath">S</span> exists such that <var class="Arg">G</var> is isomorphic to a subgroup of the automorphism group of <span class="SimpleMath">S</span> that contains all inner automorphisms of <span class="SimpleMath">S</span>.</p>

<p>Equivalently, <var class="Arg">G</var> is almost simple if and only if it has a unique minimal normal subgroup <span class="SimpleMath">N</span> and if <span class="SimpleMath">N</span> is a nonabelian simple group.</p>

<p>Note that an almost simple group is <em>not</em> defined as an extension of a simple group by outer automorphisms, since we want to exclude extensions of groups of prime order. In particular, a <em>simple</em> group is <em>almost simple</em> if and only if it is nonabelian.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsAlmostSimpleGroup( AlternatingGroup( 5 ) );</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsAlmostSimpleGroup( SymmetricGroup( 5 ) );</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsAlmostSimpleGroup( SymmetricGroup( 3 ) );</span>
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsAlmostSimpleGroup( SL( 2, 5 ) );            </span>
false
</pre></div>

<p><a id="X7C6AA6897C4409AC" name="X7C6AA6897C4409AC"></a></p>

<h5>39.15-12 <span class="Heading">IsomorphismTypeInfoFiniteSimpleGroup</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsomorphismTypeInfoFiniteSimpleGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsomorphismTypeInfoFiniteSimpleGroup</code>( <var class="Arg">n</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a finite simple group <var class="Arg">G</var>, <code class="func">IsomorphismTypeInfoFiniteSimpleGroup</code> returns a record with the components <code class="code">series</code>, <code class="code">name</code> and possibly <code class="code">parameter</code>, describing the isomorphism type of <var class="Arg">G</var>. The component <code class="code">name</code> is a string that gives name(s) for <var class="Arg">G</var>, and <code class="code">series</code> is a string that describes the following series.</p>

<p>(If different characterizations of <var class="Arg">G</var> are possible only one is given by <code class="code">series</code> and <code class="code">parameter</code>, while <code class="code">name</code> may give several names.)</p>


<dl>
<dt><strong class="Mark"><code class="code">"A"</code></strong></dt>
<dd><p>Alternating groups, <code class="code">parameter</code> gives the natural degree.</p>

</dd>
<dt><strong class="Mark"><code class="code">"L"</code></strong></dt>
<dd><p>Linear groups (Chevalley type <span class="SimpleMath">A</span>), <code class="code">parameter</code> is a list <span class="SimpleMath">[ n, q ]</span> that indicates <span class="SimpleMath">L(n,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"2A"</code></strong></dt>
<dd><p>Twisted Chevalley type <span class="SimpleMath">^2A</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">[ n, q ]</span> that indicates <span class="SimpleMath">^2A(n,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"B"</code></strong></dt>
<dd><p>Chevalley type <span class="SimpleMath">B</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">[n, q ]</span> that indicates <span class="SimpleMath">B(n,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"2B"</code></strong></dt>
<dd><p>Twisted Chevalley type <span class="SimpleMath">^2B</span>, <code class="code">parameter</code> is a value <span class="SimpleMath">q</span> that indicates <span class="SimpleMath">^2B(2,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"C"</code></strong></dt>
<dd><p>Chevalley type <span class="SimpleMath">C</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">[ n, q ]</span> that indicates <span class="SimpleMath">C(n,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"D"</code></strong></dt>
<dd><p>Chevalley type <span class="SimpleMath">D</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">[ n, q ]</span> that indicates <span class="SimpleMath">D(n,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"2D"</code></strong></dt>
<dd><p>Twisted Chevalley type <span class="SimpleMath">^2D</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">[ n, q ]</span> that indicates <span class="SimpleMath">^2D(n,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"3D"</code></strong></dt>
<dd><p>Twisted Chevalley type <span class="SimpleMath">^3D</span>, <code class="code">parameter</code> is a value <span class="SimpleMath">q</span> that indicates <span class="SimpleMath">^3D(4,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"E"</code></strong></dt>
<dd><p>Exceptional Chevalley type <span class="SimpleMath">E</span>, <code class="code">parameter</code> is a list <span class="SimpleMath">[ n, q ]</span> that indicates <span class="SimpleMath">E_n(q)</span>. The value of <var class="Arg">n</var> is 6, 7, or 8.</p>

</dd>
<dt><strong class="Mark"><code class="code">"2E"</code></strong></dt>
<dd><p>Twisted exceptional Chevalley type <span class="SimpleMath">E_6</span>, <code class="code">parameter</code> is a value <span class="SimpleMath">q</span> that indicates <span class="SimpleMath">^2E_6(q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"F"</code></strong></dt>
<dd><p>Exceptional Chevalley type <span class="SimpleMath">F</span>, <code class="code">parameter</code> is a value <span class="SimpleMath">q</span> that indicates <span class="SimpleMath">F(4,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"2F"</code></strong></dt>
<dd><p>Twisted exceptional Chevalley type <span class="SimpleMath">^2F</span> (Ree groups), <code class="code">parameter</code> is a value <span class="SimpleMath">q</span> that indicates <span class="SimpleMath">^2F(4,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"G"</code></strong></dt>
<dd><p>Exceptional Chevalley type <span class="SimpleMath">G</span>, <code class="code">parameter</code> is a value <span class="SimpleMath">q</span> that indicates <span class="SimpleMath">G(2,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"2G"</code></strong></dt>
<dd><p>Twisted exceptional Chevalley type <span class="SimpleMath">^2G</span> (Ree groups), <code class="code">parameter</code> is a value <span class="SimpleMath">q</span> that indicates <span class="SimpleMath">^2G(2,q)</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">"Spor"</code></strong></dt>
<dd><p>Sporadic simple groups, <code class="code">name</code> gives the name.</p>

</dd>
<dt><strong class="Mark"><code class="code">"Z"</code></strong></dt>
<dd><p>Cyclic groups of prime size, <code class="code">parameter</code> gives the size.</p>

</dd>
</dl>
<p>An equal sign in the name denotes different naming schemes for the same group, a tilde sign abstract isomorphisms between groups constructed in a different way.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup(</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">                            Group((4,5)(6,7),(1,2,4)(3,5,6)));</span>
rec( 
  name := "A(1,7) = L(2,7) ~ B(1,7) = O(3,7) ~ C(1,7) = S(2,7) ~ 2A(1,\
7) = U(2,7) ~ A(2,2) = L(3,2)", parameter := [ 2, 7 ], series := "L" )
</pre></div>

<p>For a positive integer <var class="Arg">n</var>, <code class="func">IsomorphismTypeInfoFiniteSimpleGroup</code> returns <code class="keyw">fail</code> if <var class="Arg">n</var> is not the order of a finite simple group, and a record as described for the case of a group <var class="Arg">G</var> otherwise. If more than one simple group of order <var class="Arg">n</var> exists then the result record contains only the <code class="code">name</code> component, a string that lists the two possible isomorphism types of simple groups of this order.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( 5 );    </span>
rec( name := "Z(5)", parameter := 5, series := "Z" )
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup( 6 );</span>
fail
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsomorphismTypeInfoFiniteSimpleGroup(Size(SymplecticGroup(6,3))/2);</span>
rec( 
  name := "cannot decide from size alone between B(3,3) = O(7,3) and C\
(3,3) = S(6,3)", parameter := [ 3, 3 ] )
</pre></div>

<p><a id="X8492B05B822AC58C" name="X8492B05B822AC58C"></a></p>

<h5>39.15-13 SimpleGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SimpleGroup</code>( <var class="Arg">id</var>[, <var class="Arg">param</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function will construct <strong class="button">an</strong> instance of the specified simple group. Groups are specified via their name in ATLAS style notation, with parameters added if necessary. The intelligence applied to parsing the name is limited, and at the moment no proper extensions can be constructed. For groups who do not have a permutation representation of small degree the ATLASREP package might need to be installed to construct theses groups.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=SimpleGroup("M(23)");</span>
M23
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(g);</span>
10200960
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=SimpleGroup("PSL",3,5);</span>
PSL(3,5)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(g);</span>
372000
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=SimpleGroup("PSp6",2);    </span>
PSp(6,2)
</pre></div>

<p><a id="X839CDD8C7AE39FD6" name="X839CDD8C7AE39FD6"></a></p>

<h5>39.15-14 SimpleGroupsIterator</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SimpleGroupsIterator</code>( [<var class="Arg">start</var>[, <var class="Arg">end</var>]] )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns an iterator that will run over all simple groups, starting at order <var class="Arg">start</var> if specified, up to order <span class="SimpleMath">10^18</span> (or -- if specified -- order <var class="Arg">end</var>). If the option <var class="Arg">NOPSL2</var> is given, groups of type <span class="SimpleMath">PSL_2(q)</span> are omitted.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">it:=SimpleGroupsIterator(20000);</span>
&lt;iterator&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List([1..8],x-&gt;NextIterator(it)); </span>
[ A8, PSL(3,4), PSL(2,37), PSp(4,3), Sz(8), PSL(2,32), PSL(2,41), 
  PSL(2,43) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">it:=SimpleGroupsIterator(1,2000);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">l:=[];;for i in it do Add(l,i);od;l;</span>
[ A5, PSL(2,7), A6, PSL(2,8), PSL(2,11), PSL(2,13) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">it:=SimpleGroupsIterator(20000,100000:NOPSL2);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">l:=[];;for i in it do Add(l,i);od;l;</span>
[ A8, PSL(3,4), PSp(4,3), Sz(8), PSU(3,4), M12 ]
</pre></div>

<p><a id="X872E93F586F54FCE" name="X872E93F586F54FCE"></a></p>

<h5>39.15-15 SmallSimpleGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SmallSimpleGroup</code>( <var class="Arg">order</var>[, <var class="Arg">i</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">i</var>th simple group of order <var class="Arg">order</var> in the stored list, given in a small-degree permutation representation, or <code class="func">fail</code> (<a href="chap20.html#X8294AAC9860E87E5"><span class="RefLink">20.2-1</span></a>) if no such simple group exists.</p>

<p>If <var class="Arg">i</var> is not given, it defaults to 1. Currently, all simple groups of order less than <span class="SimpleMath">10^6</span> are available via this function.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SmallSimpleGroup(60);</span>
A5
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SmallSimpleGroup(20160,1);</span>
A8
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SmallSimpleGroup(20160,2);</span>
PSL(3,4)
</pre></div>

<p><a id="X7EB47BF27D8CBF72" name="X7EB47BF27D8CBF72"></a></p>

<h5>39.15-16 AllSmallNonabelianSimpleGroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AllSmallNonabelianSimpleGroups</code>( <var class="Arg">orders</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A list of all nonabelian simple groups whose order lies in the range <var class="Arg">orders</var>.</p>

<p>The groups are given in small-degree permutation representations. The returned list is sorted by ascending group order. Currently, all simple groups of order less than <span class="SimpleMath">10^6</span> are available via this function.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(AllSmallNonabelianSimpleGroups([1..1000000]),</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">        StructureDescription);</span>
[ "A5", "PSL(3,2)", "A6", "PSL(2,8)", "PSL(2,11)", "PSL(2,13)", 
  "PSL(2,17)", "A7", "PSL(2,19)", "PSL(2,16)", "PSL(3,3)", 
  "PSU(3,3)", "PSL(2,23)", "PSL(2,25)", "M11", "PSL(2,27)", 
  "PSL(2,29)", "PSL(2,31)", "A8", "PSL(3,4)", "PSL(2,37)", "O(5,3)", 
  "Sz(8)", "PSL(2,32)", "PSL(2,41)", "PSL(2,43)", "PSL(2,47)", 
  "PSL(2,49)", "PSU(3,4)", "PSL(2,53)", "M12", "PSL(2,59)", 
  "PSL(2,61)", "PSU(3,5)", "PSL(2,67)", "J1", "PSL(2,71)", "A9", 
  "PSL(2,73)", "PSL(2,79)", "PSL(2,64)", "PSL(2,81)", "PSL(2,83)", 
  "PSL(2,89)", "PSL(3,5)", "M22", "PSL(2,97)", "PSL(2,101)", 
  "PSL(2,103)", "HJ", "PSL(2,107)", "PSL(2,109)", "PSL(2,113)", 
  "PSL(2,121)", "PSL(2,125)", "O(5,4)" ]
</pre></div>

<p><a id="X81E22D07871DF37E" name="X81E22D07871DF37E"></a></p>

<h5>39.15-17 IsFinitelyGeneratedGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsFinitelyGeneratedGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>tests whether the group <var class="Arg">G</var> can be generated by a finite number of generators. (This property is mainly used to obtain finiteness conditions.)</p>

<p>Note that this is a pure existence statement. Even if a group is known to be generated by a finite number of elements, it can be very hard or even impossible to obtain such a generating set if it is not known.</p>

<p><a id="X8648EDA287829755" name="X8648EDA287829755"></a></p>

<h5>39.15-18 IsSubsetLocallyFiniteGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsSubsetLocallyFiniteGroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A group is called locally finite if every finitely generated subgroup is finite. This property checks whether the group <var class="Arg">U</var> is a subset of a locally finite group. This is used to check whether finite generation will imply finiteness, as it does for example for permutation groups.</p>

<p><a id="X8089F18C810B7E3E" name="X8089F18C810B7E3E"></a></p>

<h5>39.15-19 IsPGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsPGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A <em><span class="SimpleMath">p</span>-group</em> is a finite group whose order (see <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>)) is of the form <span class="SimpleMath">p^n</span> for a prime integer <span class="SimpleMath">p</span> and a nonnegative integer <span class="SimpleMath">n</span>. <code class="func">IsPGroup</code> returns <code class="keyw">true</code> if <var class="Arg">G</var> is a <span class="SimpleMath">p</span>-group, and <code class="keyw">false</code> otherwise.</p>

<p><a id="X87356BAA7E9E2142" name="X87356BAA7E9E2142"></a></p>

<h5>39.15-20 PrimePGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PrimePGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">G</var> is a nontrivial <span class="SimpleMath">p</span>-group (see <code class="func">IsPGroup</code> (<a href="chap39.html#X8089F18C810B7E3E"><span class="RefLink">39.15-19</span></a>)), <code class="func">PrimePGroup</code> returns the prime integer <span class="SimpleMath">p</span>; if <var class="Arg">G</var> is trivial then <code class="func">PrimePGroup</code> returns <code class="keyw">fail</code>. Otherwise an error is issued.</p>

<p>(One should avoid a common error of writing <code class="code">if IsPGroup(g) then ... PrimePGroup(g) ...</code> where the code represented by dots assumes that <code class="code">PrimePGroup(g)</code> is an integer.)</p>

<p><a id="X863434AD7DDE514B" name="X863434AD7DDE514B"></a></p>

<h5>39.15-21 PClassPGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PClassPGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <span class="SimpleMath">p</span>-class of a <span class="SimpleMath">p</span>-group <var class="Arg">G</var> (see <code class="func">IsPGroup</code> (<a href="chap39.html#X8089F18C810B7E3E"><span class="RefLink">39.15-19</span></a>)) is the length of the lower <span class="SimpleMath">p</span>-central series (see <code class="func">PCentralSeries</code> (<a href="chap39.html#X7809B7ED792669F3"><span class="RefLink">39.17-13</span></a>)) of <var class="Arg">G</var>. If <var class="Arg">G</var> is not a <span class="SimpleMath">p</span>-group then an error is issued.</p>

<p><a id="X840A4F937ABF15E1" name="X840A4F937ABF15E1"></a></p>

<h5>39.15-22 RankPGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RankPGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a <span class="SimpleMath">p</span>-group <var class="Arg">G</var> (see <code class="func">IsPGroup</code> (<a href="chap39.html#X8089F18C810B7E3E"><span class="RefLink">39.15-19</span></a>)), <code class="func">RankPGroup</code> returns the <em>rank</em> of <var class="Arg">G</var>, which is defined as the minimal size of a generating system of <var class="Arg">G</var>. If <var class="Arg">G</var> is not a <span class="SimpleMath">p</span>-group then an error is issued.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h:=Group((1,2,3,4),(1,3));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">PClassPGroup(h);</span>
2
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RankPGroup(h);</span>
2
</pre></div>

<p><a id="X81130F9A7CFCF6BF" name="X81130F9A7CFCF6BF"></a></p>

<h5>39.15-23 IsPSolvable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsPSolvable</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A finite group is <span class="SimpleMath">p</span>-solvable if every chief factor either has order not divisible by <span class="SimpleMath">p</span>, or is solvable.</p>

<p><a id="X87415A8485FCF510" name="X87415A8485FCF510"></a></p>

<h5>39.15-24 IsPNilpotent</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsPNilpotent</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A group is <span class="SimpleMath">p</span>-nilpotent if it possesses a normal <span class="SimpleMath">p</span>-complement.</p>

<p><a id="X7F8264FA796B2B7D" name="X7F8264FA796B2B7D"></a></p>

<h4>39.16 <span class="Heading">Numerical Group Attributes</span></h4>

<p>This section gives only some examples of numerical group attributes, so it should not serve as a collection of all numerical group attributes. The manual contains more such attributes documented in this manual, for example, <code class="func">NrConjugacyClasses</code> (<a href="chap39.html#X8733F87B7E4C9903"><span class="RefLink">39.10-5</span></a>), <code class="func">NilpotencyClassOfGroup</code> (<a href="chap39.html#X7E3056237C6A5D43"><span class="RefLink">39.15-4</span></a>) and others.</p>

<p>Note also that some functions, such as <code class="func">EulerianFunction</code> (<a href="chap39.html#X843E0CCA8351FDF4"><span class="RefLink">39.16-3</span></a>), are mathematical attributes, but not <strong class="pkg">GAP</strong> attributes (see <a href="chap13.html#X7C701DBF7BAE649A"><span class="RefLink">13.5</span></a>) as they are depending on a parameter.</p>

<p><a id="X812827937F403300" name="X812827937F403300"></a></p>

<h5>39.16-1 AbelianInvariants</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AbelianInvariants</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the abelian invariants (also sometimes called primary decomposition) of the commutator factor group of the group <var class="Arg">G</var>. These are given as a list of prime-powers or zeroes and describe the structure of <span class="SimpleMath"><var class="Arg">G</var>/<var class="Arg">G</var>'</span> as a direct product of cyclic groups of prime power (or infinite) order.</p>

<p>(See <code class="func">IndependentGeneratorsOfAbelianGroup</code> (<a href="chap39.html#X7D1574457B152333"><span class="RefLink">39.22-5</span></a>) to obtain actual generators).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2),(5,6));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariants(g);</span>
[ 2, 2 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h:=FreeGroup(2);;h:=h/[h.1^3];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariants(h);</span>
[ 0, 3 ]
</pre></div>

<p><a id="X7D44470C7DA59C1C" name="X7D44470C7DA59C1C"></a></p>

<h5>39.16-2 Exponent</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Exponent</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The exponent <span class="SimpleMath">e</span> of a group <var class="Arg">G</var> is the lcm of the orders of its elements, that is, <span class="SimpleMath">e</span> is the smallest integer such that <span class="SimpleMath">g^e = 1</span> for all <span class="SimpleMath">g ∈ <var class="Arg">G</var></span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Exponent(g);</span>
12
</pre></div>

<p><a id="X843E0CCA8351FDF4" name="X843E0CCA8351FDF4"></a></p>

<h5>39.16-3 EulerianFunction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; EulerianFunction</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the number of <var class="Arg">n</var>-tuples <span class="SimpleMath">(g_1, g_2, ..., g_n)</span> of elements of the group <var class="Arg">G</var> that generate the whole group <var class="Arg">G</var>. The elements of such an <var class="Arg">n</var>-tuple need not be different.</p>

<p>In <a href="chapBib.html#biBHal36">[Hal36]</a>, the notation <span class="SimpleMath">ϕ_<var class="Arg">n</var>(<var class="Arg">G</var>)</span> is used for the value returned by <code class="func">EulerianFunction</code>, and the quotient of <span class="SimpleMath">ϕ_<var class="Arg">n</var>(<var class="Arg">G</var>)</span> by the order of the automorphism group of <var class="Arg">G</var> is called <span class="SimpleMath">d_<var class="Arg">n</var>(<var class="Arg">G</var>)</span>. If <var class="Arg">G</var> is a nonabelian simple group then <span class="SimpleMath">d_<var class="Arg">n</var>(<var class="Arg">G</var>)</span> is the greatest number <span class="SimpleMath">d</span> for which the direct product of <span class="SimpleMath">d</span> groups isomorphic with <var class="Arg">G</var> can be generated by <var class="Arg">n</var> elements.</p>

<p>If the Library of Tables of Marks (see Chapter <a href="chap70.html#X84DBFB8287C3F1B4"><span class="RefLink">70</span></a>) covers the group <var class="Arg">G</var>, you may also use <code class="func">EulerianFunctionByTom</code> (<a href="chap70.html#X7B1C1A7C867A4082"><span class="RefLink">70.9-9</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">EulerianFunction( g, 2 );</span>
432
</pre></div>

<p><a id="X7AEDEDF67CFED672" name="X7AEDEDF67CFED672"></a></p>

<h4>39.17 <span class="Heading">Subgroup Series</span></h4>

<p>In group theory many subgroup series are considered, and <strong class="pkg">GAP</strong> provides commands to compute them. In the following sections, there is always a series <span class="SimpleMath">G = U_1 &gt; U_2 &gt;&gt; U_m = ⟨ 1 ⟩</span> of subgroups considered. A series also may stop without reaching <span class="SimpleMath">G</span> or <span class="SimpleMath">⟨ 1 ⟩</span>.</p>

<p>A series is called <em>subnormal</em> if every <span class="SimpleMath">U_{i+1}</span> is normal in <span class="SimpleMath">U_i</span>.</p>

<p>A series is called <em>normal</em> if every <span class="SimpleMath">U_i</span> is normal in <span class="SimpleMath">G</span>.</p>

<p>A series of normal subgroups is called <em>central</em> if <span class="SimpleMath">U_i/U_{i+1}</span> is central in <span class="SimpleMath">G / U_{i+1}</span>.</p>

<p>We call a series <em>refinable</em> if intermediate subgroups can be added to the series without destroying the properties of the series.</p>

<p>Unless explicitly declared otherwise, all subgroup series are descending. That is they are stored in decreasing order.</p>

<p><a id="X7BDD116F7833800F" name="X7BDD116F7833800F"></a></p>

<h5>39.17-1 ChiefSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ChiefSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is a series of normal subgroups of <var class="Arg">G</var> which cannot be refined further. That is there is no normal subgroup <span class="SimpleMath">N</span> of <var class="Arg">G</var> with <span class="SimpleMath">U_i &gt; N &gt; U_{i+1}</span>. This attribute returns <em>one</em> chief series (of potentially many possibilities).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ChiefSeries(g);</span>
[ Group([ (1,2,3,4), (1,2) ]), 
  Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]), 
  Group([ (1,4)(2,3), (1,3)(2,4) ]), Group(()) ]
</pre></div>

<p><a id="X7AC93E977AC9ED58" name="X7AC93E977AC9ED58"></a></p>

<h5>39.17-2 ChiefSeriesThrough</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ChiefSeriesThrough</code>( <var class="Arg">G</var>, <var class="Arg">l</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is a chief series of the group <var class="Arg">G</var> going through the normal subgroups in the list <var class="Arg">l</var>, which must be a list of normal subgroups of <var class="Arg">G</var> contained in each other, sorted by descending size. This attribute returns <em>one</em> chief series (of potentially many possibilities).</p>

<p><a id="X8724E15F81B51173" name="X8724E15F81B51173"></a></p>

<h5>39.17-3 ChiefSeriesUnderAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ChiefSeriesUnderAction</code>( <var class="Arg">H</var>, <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a series of normal subgroups of <var class="Arg">G</var> which are invariant under <var class="Arg">H</var> such that the series cannot be refined any further. <var class="Arg">G</var> must be a subgroup of <var class="Arg">H</var>. This attribute returns <em>one</em> such series (of potentially many possibilities).</p>

<p><a id="X7A0E7A8B8495B79D" name="X7A0E7A8B8495B79D"></a></p>

<h5>39.17-4 SubnormalSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SubnormalSeries</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">U</var> is a subgroup of <var class="Arg">G</var> this operation returns a subnormal series that descends from <var class="Arg">G</var> to a subnormal subgroup <span class="SimpleMath">V ≥</span><var class="Arg">U</var>. If <var class="Arg">U</var> is subnormal, <span class="SimpleMath">V =</span> <var class="Arg">U</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">s:=SubnormalSeries(g,Group((1,2)(3,4)));</span>
[ Group([ (1,2,3,4), (1,2) ]), Group([ (1,2)(3,4), (1,3)(2,4) ]), 
  Group([ (1,2)(3,4) ]) ]
</pre></div>

<p><a id="X81CDCBD67BC98A5A" name="X81CDCBD67BC98A5A"></a></p>

<h5>39.17-5 CompositionSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CompositionSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A composition series is a subnormal series which cannot be refined. This attribute returns <em>one</em> composition series (of potentially many possibilities).</p>

<p><a id="X82C0D0217ACB2042" name="X82C0D0217ACB2042"></a></p>

<h5>39.17-6 DisplayCompositionSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DisplayCompositionSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Displays a composition series of <var class="Arg">G</var> in a nice way, identifying the simple factors.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CompositionSeries(g);</span>
[ Group([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]), 
  Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]), 
  Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,3)(2,4) ]), Group(()) 
 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">DisplayCompositionSeries(Group((1,2,3,4,5,6,7),(1,2)));</span>
G (2 gens, size 5040)
 | Z(2)
S (5 gens, size 2520)
 | A(7)
1 (0 gens, size 1)
</pre></div>

<p><a id="X7A879948834BD889" name="X7A879948834BD889"></a></p>

<h5>39.17-7 DerivedSeriesOfGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DerivedSeriesOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The derived series of a group is obtained by <span class="SimpleMath">U_{i+1} = U_i'</span>. It stops if <span class="SimpleMath">U_i</span> is perfect.</p>

<p><a id="X7A9AA1577CEC891F" name="X7A9AA1577CEC891F"></a></p>

<h5>39.17-8 DerivedLength</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DerivedLength</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The derived length of a group is the number of steps in the derived series. (As there is always the group, it is the series length minus 1.)</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(DerivedSeriesOfGroup(g),Size);</span>
[ 24, 12, 4, 1 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">DerivedLength(g);</span>
3
</pre></div>

<p><a id="X83F057E5791944D6" name="X83F057E5791944D6"></a></p>

<h5>39.17-9 <span class="Heading">ElementaryAbelianSeries</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ElementaryAbelianSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ElementaryAbelianSeriesLargeSteps</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ElementaryAbelianSeries</code>( <var class="Arg">list</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a series of normal subgroups of <span class="SimpleMath">G</span> such that all factors are elementary abelian. If the group is not solvable (and thus no such series exists) it returns <code class="keyw">fail</code>.</p>

<p>The variant <code class="func">ElementaryAbelianSeriesLargeSteps</code> tries to make the steps in this series large (by eliminating intermediate subgroups if possible) at a small additional cost.</p>

<p>In the third variant, an elementary abelian series through the given series of normal subgroups in the list <var class="Arg">list</var> is constructed.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(ElementaryAbelianSeries(g),Size);</span>
[ 24, 12, 4, 1 ]
</pre></div>

<p><a id="X782BD7A47D6B6503" name="X782BD7A47D6B6503"></a></p>

<h5>39.17-10 InvariantElementaryAbelianSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InvariantElementaryAbelianSeries</code>( <var class="Arg">G</var>, <var class="Arg">morph</var>[, <var class="Arg">N</var>[, <var class="Arg">fine</var>]] )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a (solvable) group <var class="Arg">G</var> and a list of automorphisms <var class="Arg">morph</var> of <var class="Arg">G</var>, this command finds a normal series of <var class="Arg">G</var> with elementary abelian factors such that every group in this series is invariant under every automorphism in <var class="Arg">morph</var>.</p>

<p>If a normal subgroup <var class="Arg">N</var> of <var class="Arg">G</var> which is invariant under <var class="Arg">morph</var> is given, this series is chosen to contain <var class="Arg">N</var>. No tests are performed to check the validity of the arguments.</p>

<p>The series obtained will be constructed to prefer large steps unless <var class="Arg">fine</var> is given as <code class="keyw">true</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,3));</span>
Group([ (1,2,3,4), (1,3) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">hom:=GroupHomomorphismByImages(g,g,GeneratorsOfGroup(g),</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[(1,4,3,2),(1,4)(2,3)]);</span>
[ (1,2,3,4), (1,3) ] -&gt; [ (1,4,3,2), (1,4)(2,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">InvariantElementaryAbelianSeries(g,[hom]);</span>
[ Group([ (1,2,3,4), (1,3) ]), Group([ (1,3)(2,4) ]), Group(()) ]
</pre></div>

<p><a id="X879D55A67DB42676" name="X879D55A67DB42676"></a></p>

<h5>39.17-11 LowerCentralSeriesOfGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LowerCentralSeriesOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The lower central series of a group <var class="Arg">G</var> is defined as <span class="SimpleMath">U_{i+1}:= [<var class="Arg">G</var>, U_i]</span>. It is a central series of normal subgroups. The name derives from the fact that <span class="SimpleMath">U_i</span> is contained in the <span class="SimpleMath">i</span>-th step subgroup of any central series.</p>

<p><a id="X8428592E8773CD7B" name="X8428592E8773CD7B"></a></p>

<h5>39.17-12 UpperCentralSeriesOfGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; UpperCentralSeriesOfGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The upper central series of a group <var class="Arg">G</var> is defined as an ending series <span class="SimpleMath">U_i / U_{i+1}:= Z(<var class="Arg">G</var>/U_{i+1})</span>. It is a central series of normal subgroups. The name derives from the fact that <span class="SimpleMath">U_i</span> contains every <span class="SimpleMath">i</span>-th step subgroup of a central series.</p>

<p><a id="X7809B7ED792669F3" name="X7809B7ED792669F3"></a></p>

<h5>39.17-13 PCentralSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PCentralSeries</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The <var class="Arg">p</var>-central series of <var class="Arg">G</var> is defined by <span class="SimpleMath">U_1:= <var class="Arg">G</var></span>, <span class="SimpleMath">U_i:= [<var class="Arg">G</var>, U_{i-1}] U_{i-1}^<var class="Arg">p</var></span>.</p>

<p><a id="X82A34BD681F24A94" name="X82A34BD681F24A94"></a></p>

<h5>39.17-14 JenningsSeries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; JenningsSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a <span class="SimpleMath">p</span>-group <var class="Arg">G</var>, this function returns its Jennings series. This series is defined by setting <span class="SimpleMath">G_1 = <var class="Arg">G</var></span> and for <span class="SimpleMath">i ≥ 0</span>, <span class="SimpleMath">G_{i+1} = [G_i,<var class="Arg">G</var>] G_j^p</span>, where <span class="SimpleMath">j</span> is the smallest integer <span class="SimpleMath">≥ i/p</span>.</p>

<p><a id="X7C08A8B77EC09CFF" name="X7C08A8B77EC09CFF"></a></p>

<h5>39.17-15 DimensionsLoewyFactors</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DimensionsLoewyFactors</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This operation computes the dimensions of the factors of the Loewy series of <var class="Arg">G</var>. (See <a href="chapBib.html#biBHup82">[HB82, p. 157]</a> for the slightly complicated definition of the Loewy Series.)</p>

<p>The dimensions are computed via the <code class="func">JenningsSeries</code> (<a href="chap39.html#X82A34BD681F24A94"><span class="RefLink">39.17-14</span></a>) without computing the Loewy series itself.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G:= SmallGroup( 3^6, 100 );</span>
&lt;pc group of size 729 with 6 generators&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">JenningsSeries( G );</span>
[ &lt;pc group of size 729 with 6 generators&gt;, Group([ f3, f4, f5, f6 ]),
  Group([ f4, f5, f6 ]), Group([ f5, f6 ]), Group([ f5, f6 ]), 
  Group([ f5, f6 ]), Group([ f6 ]), Group([ f6 ]), Group([ f6 ]), 
  Group([ &lt;identity&gt; of ... ]) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">DimensionsLoewyFactors(G);</span>
[ 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 
  27, 27, 27, 27, 27, 27, 27, 27, 27, 26, 25, 23, 22, 20, 19, 17, 16, 
  14, 13, 11, 10, 8, 7, 5, 4, 2, 1 ]
</pre></div>

<p><a id="X84112774812180DD" name="X84112774812180DD"></a></p>

<h5>39.17-16 AscendingChain</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AscendingChain</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function computes an ascending chain of subgroups from <var class="Arg">U</var> to <var class="Arg">G</var>. This chain is given as a list whose first entry is <var class="Arg">U</var> and the last entry is <var class="Arg">G</var>. The function tries to make the links in this chain small.</p>

<p>The option <code class="code">refineIndex</code> can be used to give a bound for refinements of steps to avoid <strong class="pkg">GAP</strong> trying to enforce too small steps. The option <code class="code">cheap</code> (if set to <code class="keyw">true</code>) will overall limit the amount of heuristic searches.</p>

<p><a id="X7C5029EE86D7FC96" name="X7C5029EE86D7FC96"></a></p>

<h5>39.17-17 IntermediateGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IntermediateGroup</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This routine tries to find a subgroup <span class="SimpleMath">E</span> of <var class="Arg">G</var>, such that <span class="SimpleMath"><var class="Arg">G</var> &gt; E &gt; <var class="Arg">U</var></span> holds. If <var class="Arg">U</var> is maximal in <var class="Arg">G</var>, the function returns <code class="keyw">fail</code>. This is done by finding minimal blocks for the operation of <var class="Arg">G</var> on the right cosets of <var class="Arg">U</var>.</p>

<p><a id="X781661FB78DC83B5" name="X781661FB78DC83B5"></a></p>

<h5>39.17-18 IntermediateSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IntermediateSubgroups</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a list of all subgroups of <var class="Arg">G</var> that properly contain <var class="Arg">U</var>; that is all subgroups between <var class="Arg">G</var> and <var class="Arg">U</var>. It returns a record with a component <code class="code">subgroups</code>, which is a list of these subgroups, as well as a component <code class="code">inclusions</code>, which lists all maximality inclusions among these subgroups. A maximality inclusion is given as a list <span class="SimpleMath">[i, j]</span> indicating that the subgroup number <span class="SimpleMath">i</span> is a maximal subgroup of the subgroup number <span class="SimpleMath">j</span>, the numbers <span class="SimpleMath">0</span> and <span class="SimpleMath">1 +</span> <code class="code">Length(subgroups)</code> are used to denote <var class="Arg">U</var> and <var class="Arg">G</var>, respectively.</p>

<p><a id="X84091B0A7E401E2B" name="X84091B0A7E401E2B"></a></p>

<h4>39.18 <span class="Heading">Factor Groups</span></h4>

<p><a id="X80FC390C7F38A13F" name="X80FC390C7F38A13F"></a></p>

<h5>39.18-1 NaturalHomomorphismByNormalSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NaturalHomomorphismByNormalSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NaturalHomomorphismByNormalSubgroupNC</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a homomorphism from <var class="Arg">G</var> to another group whose kernel is <var class="Arg">N</var>. <strong class="pkg">GAP</strong> will try to select the image group as to make computations in it as efficient as possible. As the factor group <span class="SimpleMath"><var class="Arg">G</var>/<var class="Arg">N</var></span> can be identified with the image of <var class="Arg">G</var> this permits efficient computations in the factor group. The homomorphism returned is not necessarily surjective, so <code class="func">ImagesSource</code> (<a href="chap32.html#X7D23C1CE863DACD8"><span class="RefLink">32.4-1</span></a>) should be used instead of <code class="func">Range</code> (<a href="chap32.html#X7B6FD7277CDE9FCB"><span class="RefLink">32.3-7</span></a>) to get a group isomorphic to the factor group. The <code class="code">NC</code> variant does not check whether <var class="Arg">N</var> is normal in <var class="Arg">G</var>.</p>

<p><a id="X7E6EED0185B27C48" name="X7E6EED0185B27C48"></a></p>

<h5>39.18-2 FactorGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FactorGroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FactorGroupNC</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the image of the <code class="code">NaturalHomomorphismByNormalSubgroup(<var class="Arg">G</var>,<var class="Arg">N</var>)</code>. The homomorphism will be returned by calling the function <code class="code">NaturalHomomorphism</code> on the result. The <code class="code">NC</code> version does not test whether <var class="Arg">N</var> is normal in <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;n:=Subgroup(g,[(1,2)(3,4),(1,3)(2,4)]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">hom:=NaturalHomomorphismByNormalSubgroup(g,n);</span>
[ (1,2,3,4), (1,2) ] -&gt; [ f1*f2, f1 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(ImagesSource(hom));</span>
6
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">FactorGroup(g,n);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">StructureDescription(last);</span>
"S3"
</pre></div>

<p><a id="X7816FA867BF1B8ED" name="X7816FA867BF1B8ED"></a></p>

<h5>39.18-3 CommutatorFactorGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CommutatorFactorGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the commutator factor group <span class="SimpleMath"><var class="Arg">G</var>/<var class="Arg">G</var>'</span> of the group <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CommutatorFactorGroup(g);</span>
Group([ f1 ])
</pre></div>

<p><a id="X7BB93B9778C5A0B2" name="X7BB93B9778C5A0B2"></a></p>

<h5>39.18-4 MaximalAbelianQuotient</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MaximalAbelianQuotient</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an epimorphism from <var class="Arg">G</var> onto the maximal abelian quotient of <var class="Arg">G</var>. The kernel of this epimorphism is the derived subgroup of <var class="Arg">G</var>, see <code class="func">DerivedSubgroup</code> (<a href="chap39.html#X7CC17CF179ED7EF2"><span class="RefLink">39.12-3</span></a>).</p>

<p><a id="X7FC83E4C783572E7" name="X7FC83E4C783572E7"></a></p>

<h5>39.18-5 HasAbelianFactorGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; HasAbelianFactorGroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>tests whether <var class="Arg">G</var> <span class="SimpleMath">/</span> <var class="Arg">N</var> is abelian (without explicitly constructing the factor group and without testing whether <var class="Arg">N</var> is in fact a normal subgroup).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HasAbelianFactorGroup(g,n);</span>
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HasAbelianFactorGroup(DerivedSubgroup(g),n);</span>
true
</pre></div>

<p><a id="X7FAC018680B766B7" name="X7FAC018680B766B7"></a></p>

<h5>39.18-6 HasElementaryAbelianFactorGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; HasElementaryAbelianFactorGroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>tests whether <var class="Arg">G</var> <span class="SimpleMath">/</span> <var class="Arg">N</var> is elementary abelian (without explicitly constructing the factor group and without testing whether <var class="Arg">N</var> is in fact a normal subgroup).</p>

<p><a id="X822A3AB27919BC1E" name="X822A3AB27919BC1E"></a></p>

<h5>39.18-7 CentralizerModulo</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CentralizerModulo</code>( <var class="Arg">G</var>, <var class="Arg">N</var>, <var class="Arg">elm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Computes the full preimage of the centralizer <span class="SimpleMath">C_{<var class="Arg">G</var>/<var class="Arg">N</var>}(<var class="Arg">elm</var><var class="Arg">N</var>)</span> in <var class="Arg">G</var> (without necessarily constructing the factor group).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CentralizerModulo(g,n,(1,2));</span>
Group([ (3,4), (1,3)(2,4), (1,4)(2,3) ])
</pre></div>

<p><a id="X7D8EFB2F85AA24EE" name="X7D8EFB2F85AA24EE"></a></p>

<h4>39.19 <span class="Heading">Sets of Subgroups</span></h4>

<p><a id="X7DDE67C67E871336" name="X7DDE67C67E871336"></a></p>

<h5>39.19-1 ConjugacyClassSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugacyClassSubgroups</code>( <var class="Arg">G</var>, <var class="Arg">U</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>generates the conjugacy class of subgroups of <var class="Arg">G</var> with representative <var class="Arg">U</var>. This class is an external set, so functions such as <code class="func">Representative</code> (<a href="chap30.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>), (which returns <var class="Arg">U</var>), <code class="func">ActingDomain</code> (<a href="chap41.html#X7B9DB15D80CE28B4"><span class="RefLink">41.12-3</span></a>) (which returns <var class="Arg">G</var>), <code class="func">StabilizerOfExternalSet</code> (<a href="chap41.html#X7BAFF02B7D6DF9F2"><span class="RefLink">41.12-10</span></a>) (which returns the normalizer of <var class="Arg">U</var>), and <code class="func">AsList</code> (<a href="chap30.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) work for it.</p>

<p>(The use of the <code class="code">[]</code> list access to select elements of the class is considered obsolescent and will be removed in future versions. Use <code class="func">ClassElementLattice</code> (<a href="chap39.html#X78928A3582882BFD"><span class="RefLink">39.20-2</span></a>) instead.)</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;IsNaturalSymmetricGroup(g);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">cl:=ConjugacyClassSubgroups(g,Subgroup(g,[(1,2)]));</span>
Group( [ (1,2) ] )^G
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(cl);</span>
6
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ClassElementLattice(cl,4);</span>
Group([ (2,3) ])
</pre></div>

<p><a id="X7C5BBF487977B8CD" name="X7C5BBF487977B8CD"></a></p>

<h5>39.19-2 IsConjugacyClassSubgroupsRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsConjugacyClassSubgroupsRep</code>( <var class="Arg">obj</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsConjugacyClassSubgroupsByStabilizerRep</code>( <var class="Arg">obj</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Is the representation <strong class="pkg">GAP</strong> uses for conjugacy classes of subgroups. It can be used to check whether an object is a class of subgroups. The second representation <code class="func">IsConjugacyClassSubgroupsByStabilizerRep</code> in addition is an external orbit by stabilizer and will compute its elements via a transversal of the stabilizer.</p>

<p><a id="X7E986BF48393113A" name="X7E986BF48393113A"></a></p>

<h5>39.19-3 ConjugacyClassesSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugacyClassesSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute returns a list of all conjugacy classes of subgroups of the group <var class="Arg">G</var>. It also is applicable for lattices of subgroups (see <code class="func">LatticeSubgroups</code> (<a href="chap39.html#X7B104E2C86166188"><span class="RefLink">39.20-1</span></a>)). The order in which the classes are listed depends on the method chosen by <strong class="pkg">GAP</strong>. For each class of subgroups, a representative can be accessed using <code class="func">Representative</code> (<a href="chap30.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ConjugacyClassesSubgroups(g);</span>
[ Group( () )^G, Group( [ (1,3)(2,4) ] )^G, Group( [ (3,4) ] )^G, 
  Group( [ (2,4,3) ] )^G, Group( [ (1,4)(2,3), (1,3)(2,4) ] )^G, 
  Group( [ (3,4), (1,2)(3,4) ] )^G, 
  Group( [ (1,3,2,4), (1,2)(3,4) ] )^G, Group( [ (3,4), (2,4,3) ] )^G,
  Group( [ (1,4)(2,3), (1,3)(2,4), (3,4) ] )^G, 
  Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3) ] )^G, 
  Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ] )^G ]
</pre></div>

<p><a id="X8486C25380853F9B" name="X8486C25380853F9B"></a></p>

<h5>39.19-4 ConjugacyClassesMaximalSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugacyClassesMaximalSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the conjugacy classes of maximal subgroups of <var class="Arg">G</var>. Representatives of the classes can be computed directly by <code class="func">MaximalSubgroupClassReps</code> (<a href="chap39.html#X798BF55C837DB188"><span class="RefLink">39.19-6</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ConjugacyClassesMaximalSubgroups(g);</span>
[ AlternatingGroup( [ 1 .. 4 ] )^G, Group( [ (1,2,3), (1,2) ] )^G, 
  Group( [ (1,2), (3,4), (1,3)(2,4) ] )^G ]
</pre></div>

<p><a id="X80399CD4870FFC4B" name="X80399CD4870FFC4B"></a></p>

<h5>39.19-5 AllSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AllSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a finite group <var class="Arg">G</var> <code class="func">AllSubgroups</code> returns a list of all subgroups of <var class="Arg">G</var>, intended primarily for use in class for small examples. This list will quickly get very long and in general use of <code class="func">ConjugacyClassesSubgroups</code> (<a href="chap39.html#X7E986BF48393113A"><span class="RefLink">39.19-3</span></a>) is recommended.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AllSubgroups(SymmetricGroup(3));</span>
[ Group(()), Group([ (2,3) ]), Group([ (1,2) ]), Group([ (1,3) ]), 
  Group([ (1,2,3) ]), Group([ (1,2,3), (2,3) ]) ]
</pre></div>

<p><a id="X798BF55C837DB188" name="X798BF55C837DB188"></a></p>

<h5>39.19-6 MaximalSubgroupClassReps</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MaximalSubgroupClassReps</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of conjugacy representatives of the maximal subgroups of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">MaximalSubgroupClassReps(g);</span>
[ Alt( [ 1 .. 4 ] ), Group([ (1,2,3), (1,2) ]), 
  Group([ (1,2), (3,4), (1,3)(2,4) ]) ]
</pre></div>

<p><a id="X861CD8DA790D81C2" name="X861CD8DA790D81C2"></a></p>

<h5>39.19-7 MaximalSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MaximalSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of all maximal subgroups of <var class="Arg">G</var>. This may take up much space, therefore the command should be avoided if possible. See <code class="func">ConjugacyClassesMaximalSubgroups</code> (<a href="chap39.html#X8486C25380853F9B"><span class="RefLink">39.19-4</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">MaximalSubgroups(Group((1,2,3),(1,2)));</span>
[ Group([ (1,2,3) ]), Group([ (2,3) ]), Group([ (1,2) ]), 
  Group([ (1,3) ]) ]
</pre></div>

<p><a id="X80237A847E24E6CF" name="X80237A847E24E6CF"></a></p>

<h5>39.19-8 NormalSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NormalSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of all normal subgroups of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=SymmetricGroup(4);;NormalSubgroups(g);</span>
[ Sym( [ 1 .. 4 ] ), Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]), 
  Group([ (1,4)(2,3), (1,3)(2,4) ]), Group(()) ]
</pre></div>

<p>The algorithm for the computation of normal subgroups is described in <a href="chapBib.html#biBHulpke98">[Hul98]</a>.</p>

<p><a id="X82ECAA427C987318" name="X82ECAA427C987318"></a></p>

<h5>39.19-9 MaximalNormalSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MaximalNormalSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is a list containing those proper normal subgroups of the group <var class="Arg">G</var> that are maximal among the proper normal subgroups.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">MaximalNormalSubgroups( g );</span>
[ Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) ]
</pre></div>

<p><a id="X86FDD9BA819F5644" name="X86FDD9BA819F5644"></a></p>

<h5>39.19-10 MinimalNormalSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MinimalNormalSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is a list containing those nontrivial normal subgroups of the group <var class="Arg">G</var> that are minimal among the nontrivial normal subgroups.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">MinimalNormalSubgroups( g );</span>
[ Group([ (1,4)(2,3), (1,3)(2,4) ]) ]
</pre></div>

<p><a id="X7FA267497CFC0550" name="X7FA267497CFC0550"></a></p>

<h4>39.20 <span class="Heading">Subgroup Lattice</span></h4>

<p><a id="X7B104E2C86166188" name="X7B104E2C86166188"></a></p>

<h5>39.20-1 LatticeSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LatticeSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the lattice of subgroups of the group <var class="Arg">G</var>. This lattice has the conjugacy classes of subgroups as attribute <code class="func">ConjugacyClassesSubgroups</code> (<a href="chap39.html#X7E986BF48393113A"><span class="RefLink">39.19-3</span></a>) and permits one to test maximality/minimality relations.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">l:=LatticeSubgroups(g);</span>
&lt;subgroup lattice of Sym( [ 1 .. 4 ] ), 11 classes, 30 subgroups&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ConjugacyClassesSubgroups(l);</span>
[ Group( () )^G, Group( [ (1,3)(2,4) ] )^G, Group( [ (3,4) ] )^G, 
  Group( [ (2,4,3) ] )^G, Group( [ (1,4)(2,3), (1,3)(2,4) ] )^G, 
  Group( [ (3,4), (1,2)(3,4) ] )^G, 
  Group( [ (1,3,2,4), (1,2)(3,4) ] )^G, Group( [ (3,4), (2,4,3) ] )^G,
  Group( [ (1,4)(2,3), (1,3)(2,4), (3,4) ] )^G, 
  Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3) ] )^G, 
  Group( [ (1,4)(2,3), (1,3)(2,4), (2,4,3), (3,4) ] )^G ]
</pre></div>

<p><a id="X78928A3582882BFD" name="X78928A3582882BFD"></a></p>

<h5>39.20-2 ClassElementLattice</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ClassElementLattice</code>( <var class="Arg">C</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For a class <var class="Arg">C</var> of subgroups, obtained by a lattice computation, this operation returns the <var class="Arg">n</var>-th conjugate subgroup in the class.</p>

<p><em>Because of other methods installed, calling <code class="func">AsList</code> (<a href="chap30.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) with <var class="Arg">C</var> can give a different arrangement of the class elements!</em></p>

<p>The <strong class="pkg">GAP</strong> package <strong class="pkg">XGAP</strong> permits a graphical display of the lattice of subgroups in a nice way.</p>

<p><a id="X7E5DF287825EE7BA" name="X7E5DF287825EE7BA"></a></p>

<h5>39.20-3 DotFileLatticeSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DotFileLatticeSubgroups</code>( <var class="Arg">L</var>, <var class="Arg">file</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function produces a graphical representation of the subgroup lattice <var class="Arg">L</var> in file <var class="Arg">file</var>. The output is in <code class="code">.dot</code> (also known as <code class="code">GraphViz</code> format). For details on the format, and information about how to display or edit this format see <span class="URL"><a href="http://www.graphviz.org">http://www.graphviz.org</a></span>. (On the Macintosh, the program <code class="code">OmniGraffle</code> is also able to read this format.)</p>

<p>Subgroups are labelled in the form <code class="code"><var class="Arg">i</var>-<var class="Arg">j</var></code> where <var class="Arg">i</var> is the number of the class of subgroups and <var class="Arg">j</var> the number within this class. Normal subgroups are represented by a box.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">DotFileLatticeSubgroups(l,"s4lat.dot");</span>
</pre></div>

<p><a id="X815CDA447C5DB285" name="X815CDA447C5DB285"></a></p>

<h5>39.20-4 MaximalSubgroupsLattice</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MaximalSubgroupsLattice</code>( <var class="Arg">lat</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a lattice <var class="Arg">lat</var> of subgroups this attribute contains the maximal subgroup relations among the subgroups of the lattice. It is a list corresponding to the <code class="func">ConjugacyClassesSubgroups</code> (<a href="chap39.html#X7E986BF48393113A"><span class="RefLink">39.19-3</span></a>) value of the lattice, each entry giving a list of the maximal subgroups of the representative of this class. Every maximal subgroup is indicated by a list of the form <span class="SimpleMath">[ c, n ]</span> which means that the <span class="SimpleMath">n</span>-th subgroup in class number <span class="SimpleMath">c</span> is a maximal subgroup of the representative.</p>

<p>The number <span class="SimpleMath">n</span> corresponds to access via <code class="func">ClassElementLattice</code> (<a href="chap39.html#X78928A3582882BFD"><span class="RefLink">39.20-2</span></a>) and <em>not</em> necessarily the <code class="func">AsList</code> (<a href="chap30.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) arrangement! See also <code class="func">MinimalSupergroupsLattice</code> (<a href="chap39.html#X8138997C871EDF96"><span class="RefLink">39.20-5</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">MaximalSubgroupsLattice(l);</span>
[ [  ], [ [ 1, 1 ] ], [ [ 1, 1 ] ], [ [ 1, 1 ] ], 
  [ [ 2, 1 ], [ 2, 2 ], [ 2, 3 ] ], [ [ 3, 1 ], [ 3, 6 ], [ 2, 3 ] ], 
  [ [ 2, 3 ] ], [ [ 4, 1 ], [ 3, 1 ], [ 3, 2 ], [ 3, 3 ] ], 
  [ [ 7, 1 ], [ 6, 1 ], [ 5, 1 ] ], 
  [ [ 5, 1 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ], [ 4, 4 ] ], 
  [ [ 10, 1 ], [ 9, 1 ], [ 9, 2 ], [ 9, 3 ], [ 8, 1 ], [ 8, 2 ], 
      [ 8, 3 ], [ 8, 4 ] ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">last[6];</span>
[ [ 3, 1 ], [ 3, 6 ], [ 2, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u1:=Representative(ConjugacyClassesSubgroups(l)[6]);</span>
Group([ (3,4), (1,2)(3,4) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u2:=ClassElementLattice(ConjugacyClassesSubgroups(l)[3],1);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u3:=ClassElementLattice(ConjugacyClassesSubgroups(l)[3],6);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u4:=ClassElementLattice(ConjugacyClassesSubgroups(l)[2],3);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsSubgroup(u1,u2);IsSubgroup(u1,u3);IsSubgroup(u1,u4);</span>
true
true
true
</pre></div>

<p><a id="X8138997C871EDF96" name="X8138997C871EDF96"></a></p>

<h5>39.20-5 MinimalSupergroupsLattice</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MinimalSupergroupsLattice</code>( <var class="Arg">lat</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a lattice <var class="Arg">lat</var> of subgroups this attribute contains the minimal supergroup relations among the subgroups of the lattice. It is a list corresponding to the <code class="func">ConjugacyClassesSubgroups</code> (<a href="chap39.html#X7E986BF48393113A"><span class="RefLink">39.19-3</span></a>) value of the lattice, each entry giving a list of the minimal supergroups of the representative of this class. Every minimal supergroup is indicated by a list of the form <span class="SimpleMath">[ c, n ]</span>, which means that the <span class="SimpleMath">n</span>-th subgroup in class number <span class="SimpleMath">c</span> is a minimal supergroup of the representative.</p>

<p>The number <span class="SimpleMath">n</span> corresponds to access via <code class="func">ClassElementLattice</code> (<a href="chap39.html#X78928A3582882BFD"><span class="RefLink">39.20-2</span></a>) and <em>not</em> necessarily the <code class="func">AsList</code> (<a href="chap30.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) arrangement! See also <code class="func">MaximalSubgroupsLattice</code> (<a href="chap39.html#X815CDA447C5DB285"><span class="RefLink">39.20-4</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">MinimalSupergroupsLattice(l);</span>
[ [ [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ], [ 3, 2 ], [ 3, 3 ], 
      [ 3, 4 ], [ 3, 5 ], [ 3, 6 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ], 
      [ 4, 4 ] ], [ [ 5, 1 ], [ 6, 2 ], [ 7, 2 ] ], 
  [ [ 6, 1 ], [ 8, 1 ], [ 8, 3 ] ], [ [ 8, 1 ], [ 10, 1 ] ], 
  [ [ 9, 1 ], [ 9, 2 ], [ 9, 3 ], [ 10, 1 ] ], [ [ 9, 1 ] ], 
  [ [ 9, 1 ] ], [ [ 11, 1 ] ], [ [ 11, 1 ] ], [ [ 11, 1 ] ], [  ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">last[3];</span>
[ [ 6, 1 ], [ 8, 1 ], [ 8, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u5:=ClassElementLattice(ConjugacyClassesSubgroups(l)[8],1);</span>
Group([ (3,4), (2,4,3) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u6:=ClassElementLattice(ConjugacyClassesSubgroups(l)[8],3);</span>
Group([ (1,3), (1,3,4) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsSubgroup(u5,u2);</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsSubgroup(u6,u2);</span>
true
</pre></div>

<p><a id="X7BA3484E7AE0A0E1" name="X7BA3484E7AE0A0E1"></a></p>

<h5>39.20-6 RepresentativesPerfectSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RepresentativesPerfectSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RepresentativesSimpleSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of conjugacy representatives of perfect (respectively simple) subgroups of <var class="Arg">G</var>. This uses the library of perfect groups (see <code class="func">PerfectGroup</code> (<a href="chap50.html#X7906BBA7818E9415"><span class="RefLink">50.8-2</span></a>)), thus it will issue an error if the library is insufficient to determine all perfect subgroups.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">m11:=TransitiveGroup(11,6);</span>
M(11)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">r:=RepresentativesPerfectSubgroups(m11);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(r,Size);</span>
[ 60, 60, 360, 660, 7920, 1 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(r,StructureDescription);</span>
[ "A5", "A5", "A6", "PSL(2,11)", "M11", "1" ]
</pre></div>

<p><a id="X7B2233D180DF77A1" name="X7B2233D180DF77A1"></a></p>

<h5>39.20-7 ConjugacyClassesPerfectSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugacyClassesPerfectSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of the conjugacy classes of perfect subgroups of <var class="Arg">G</var>. (see <code class="func">RepresentativesPerfectSubgroups</code> (<a href="chap39.html#X7BA3484E7AE0A0E1"><span class="RefLink">39.20-6</span></a>).)</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">r := ConjugacyClassesPerfectSubgroups(m11);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(r, x -&gt; StructureDescription(Representative(x)));</span>
[ "A5", "A5", "A6", "PSL(2,11)", "M11", "1" ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SortedList( List(r,Size) );</span>
[ 1, 1, 11, 12, 66, 132 ]
</pre></div>

<p><a id="X7BFE573187B4BEF8" name="X7BFE573187B4BEF8"></a></p>

<h5>39.20-8 Zuppos</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Zuppos</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <em>Zuppos</em> of a group are the cyclic subgroups of prime power order. (The name "Zuppo" derives from the German abbreviation for "zyklische Untergruppen von Primzahlpotenzordnung".) This attribute gives generators of all such subgroups of a group <var class="Arg">G</var>. That is all elements of <var class="Arg">G</var> of prime power order up to the equivalence that they generate the same cyclic subgroup.</p>

<p><a id="X82C12E2C81963B23" name="X82C12E2C81963B23"></a></p>

<h5>39.20-9 InfoLattice</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InfoLattice</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>is the information class used by the cyclic extension methods for subgroup lattice calculations.</p>

<p><a id="X85E613D57F28AEFF" name="X85E613D57F28AEFF"></a></p>

<h4>39.21 <span class="Heading">Specific Methods for Subgroup Lattice Computations</span></h4>

<p><a id="X86462A567DDBA6BC" name="X86462A567DDBA6BC"></a></p>

<h5>39.21-1 LatticeByCyclicExtension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LatticeByCyclicExtension</code>( <var class="Arg">G</var>[, <var class="Arg">func</var>[, <var class="Arg">noperf</var>]] )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the lattice of <var class="Arg">G</var> using the cyclic extension algorithm. If the function <var class="Arg">func</var> is given, the algorithm will discard all subgroups not fulfilling <var class="Arg">func</var> (and will also not extend them), returning a partial lattice. This can be useful to compute only subgroups with certain properties. Note however that this will <em>not</em> necessarily yield all subgroups that fulfill <var class="Arg">func</var>, but the subgroups whose subgroups are used for the construction must also fulfill <var class="Arg">func</var> as well. (In fact the filter <var class="Arg">func</var> will simply discard subgroups in the cyclic extension algorithm. Therefore the trivial subgroup will always be included.) Also note, that for such a partial lattice maximality/minimality inclusion relations cannot be computed. (If <var class="Arg">func</var> is a list of length 2, its first entry is such a discarding function, the second a function for discarding zuppos.)</p>

<p>The cyclic extension algorithm requires the perfect subgroups of <var class="Arg">G</var>. However <strong class="pkg">GAP</strong> cannot analyze the function <var class="Arg">func</var> for its implication but can only apply it. If it is known that <var class="Arg">func</var> implies solvability, the computation of the perfect subgroups can be avoided by giving a third parameter <var class="Arg">noperf</var> set to <code class="keyw">true</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=WreathProduct(Group((1,2,3),(1,2)),Group((1,2,3,4)));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">l:=LatticeByCyclicExtension(g,function(G)</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">return Size(G) in [1,2,3,6];end);</span>
&lt;subgroup lattice of &lt;permutation group of size 5184 with 
9 generators&gt;, 47 classes, 
2628 subgroups, restricted under further condition l!.func&gt;
</pre></div>

<p>The total number of classes in this example is much bigger, as the following example shows:</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LatticeSubgroups(g);</span>
&lt;subgroup lattice of &lt;permutation group of size 5184 with 
9 generators&gt;, 566 classes, 27134 subgroups&gt;
</pre></div>

<p>##</p>

<p><a id="X78918D83835A0EDF" name="X78918D83835A0EDF"></a></p>

<h5>39.21-2 InvariantSubgroupsElementaryAbelianGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InvariantSubgroupsElementaryAbelianGroup</code>( <var class="Arg">G</var>, <var class="Arg">homs</var>[, <var class="Arg">dims</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be an elementary abelian group and <var class="Arg">homs</var> be a set of automorphisms of <var class="Arg">G</var>. Then this function computes all subspaces of <var class="Arg">G</var> which are invariant under all automorphisms in <var class="Arg">homs</var>. When considering <var class="Arg">G</var> as a module for the algebra generated by <var class="Arg">homs</var>, these are all submodules. If <var class="Arg">homs</var> is empty, it computes all subgroups. If the optional parameter <var class="Arg">dims</var> is given, only submodules of this dimension are computed.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3),(4,5,6),(7,8,9));</span>
Group([ (1,2,3), (4,5,6), (7,8,9) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">hom:=GroupHomomorphismByImages(g,g,[(1,2,3),(4,5,6),(7,8,9)],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[(7,8,9),(1,2,3),(4,5,6)]);</span>
[ (1,2,3), (4,5,6), (7,8,9) ] -&gt; [ (7,8,9), (1,2,3), (4,5,6) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u:=InvariantSubgroupsElementaryAbelianGroup(g,[hom]);</span>
[ Group(()), Group([ (1,2,3)(4,5,6)(7,8,9) ]), 
  Group([ (1,3,2)(7,8,9), (1,3,2)(4,5,6) ]), 
  Group([ (7,8,9), (4,5,6), (1,2,3) ]) ]
</pre></div>

<p><a id="X7AD7804A803910AC" name="X7AD7804A803910AC"></a></p>

<h5>39.21-3 SubgroupsSolvableGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SubgroupsSolvableGroup</code>( <var class="Arg">G</var>[, <var class="Arg">opt</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function (implementing the algorithm published in <a href="chapBib.html#biBHulpke99">[Hul99]</a>) computes subgroups of a solvable group <var class="Arg">G</var>, using the homomorphism principle. It returns a list of representatives up to <var class="Arg">G</var>-conjugacy.</p>

<p>The optional argument <var class="Arg">opt</var> is a record, which may be used to put restrictions on the subgroups computed. The following record components of <var class="Arg">opt</var> are recognized and have the following effects:</p>


<dl>
<dt><strong class="Mark"><code class="code">actions</code></strong></dt>
<dd><p>must be a list of automorphisms of <var class="Arg">G</var>. If given, only groups which are invariant under all these automorphisms are computed. The algorithm must know the normalizer in <var class="Arg">G</var> of the group generated by <code class="code">actions</code> (defined formally by embedding in the semidirect product of <var class="Arg">G</var> with <var class="Arg">actions</var>). This can be given in the component <code class="code">funcnorm</code> and will be computed if this component is not given.</p>

</dd>
<dt><strong class="Mark"><code class="code">normal</code></strong></dt>
<dd><p>if set to <code class="keyw">true</code> only normal subgroups are guaranteed to be returned (though some of the returned subgroups might still be not normal).</p>

</dd>
<dt><strong class="Mark"><code class="code">consider</code></strong></dt>
<dd><p>a function to restrict the groups computed. This must be a function of five parameters, <span class="SimpleMath">C</span>, <span class="SimpleMath">A</span>, <span class="SimpleMath">N</span>, <span class="SimpleMath">B</span>, <span class="SimpleMath">M</span>, that are interpreted as follows: The arguments are subgroups of a factor <span class="SimpleMath">F</span> of <var class="Arg">G</var> in the relation <span class="SimpleMath">F ≥ C &gt; A &gt; N &gt; B &gt; M</span>. <span class="SimpleMath">N</span> and <span class="SimpleMath">M</span> are normal subgroups. <span class="SimpleMath">C</span> is the full preimage of the normalizer of <span class="SimpleMath">A/N</span> in <span class="SimpleMath">F/N</span>. When computing modulo <span class="SimpleMath">M</span> and looking for subgroups <span class="SimpleMath">U</span> such that <span class="SimpleMath">U ∩ N = B</span> and <span class="SimpleMath">⟨ U, N ⟩ = A</span>, this function is called. If it returns <code class="keyw">false</code> then all potential groups <span class="SimpleMath">U</span> (and therefore all groups later arising from them) are disregarded. This can be used for example to compute only subgroups of certain sizes.</p>

<p>(<em>This is just a restriction to speed up computations. The function may still return (invariant) subgroups which don't fulfill this condition!</em>) This parameter is used to permit calculations of some subgroups if the set of all subgroups would be too large to handle.</p>

<p>The actual groups <span class="SimpleMath">C</span>, <span class="SimpleMath">A</span>, <span class="SimpleMath">N</span> and <span class="SimpleMath">B</span> which are passed to this function are not necessarily subgroups of <var class="Arg">G</var> but might be subgroups of a proper factor group <span class="SimpleMath">F = <var class="Arg">G</var>/H</span>. Therefore the <code class="code">consider</code> function may not relate the parameter groups to <var class="Arg">G</var>.</p>

</dd>
<dt><strong class="Mark"><code class="code">retnorm</code></strong></dt>
<dd><p>if set to <code class="keyw">true</code> the function not only returns a list <code class="code">subs</code> of subgroups but also a corresponding list <code class="code">norms</code> of normalizers in the form <code class="code">[ subs, norms ]</code>.</p>

</dd>
<dt><strong class="Mark"><code class="code">series</code></strong></dt>
<dd><p>is an elementary abelian series of <var class="Arg">G</var> which will be used for the computation.</p>

</dd>
<dt><strong class="Mark"><code class="code">groups</code></strong></dt>
<dd><p>is a list of groups to seed the calculation. Only subgroups of these groups are constructed.</p>

</dd>
</dl>

<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3),(1,2),(4,5,6),(4,5),(7,8,9),(7,8));</span>
Group([ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">hom:=GroupHomomorphismByImages(g,g,</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[(1,2,3),(1,2),(4,5,6),(4,5),(7,8,9),(7,8)],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[(4,5,6),(4,5),(7,8,9),(7,8),(1,2,3),(1,2)]);</span>
[ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8) ] -&gt; 
[ (4,5,6), (4,5), (7,8,9), (7,8), (1,2,3), (1,2) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">l:=SubgroupsSolvableGroup(g,rec(actions:=[hom]));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(l,Size);</span>
[ 1, 3, 9, 27, 54, 2, 6, 18, 108, 4, 216, 8 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Length(ConjugacyClassesSubgroups(g)); # to compare</span>
162
</pre></div>

<p><a id="X7F60BBB8874DFE40" name="X7F60BBB8874DFE40"></a></p>

<h5>39.21-4 SizeConsiderFunction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SizeConsiderFunction</code>( <var class="Arg">size</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a function <code class="code">consider</code> of four arguments that can be used in <code class="func">SubgroupsSolvableGroup</code> (<a href="chap39.html#X7AD7804A803910AC"><span class="RefLink">39.21-3</span></a>) for the option <code class="code">consider</code> to compute subgroups whose sizes are divisible by <var class="Arg">size</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">l:=SubgroupsSolvableGroup(g,rec(actions:=[hom],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">consider:=SizeConsiderFunction(6)));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(l,Size);</span>
[ 1, 3, 9, 27, 54, 6, 18, 108, 216 ]
</pre></div>

<p>This example shows that in general the <code class="code">consider</code> function does not provide a perfect filter. It is guaranteed that all subgroups fulfilling the condition are returned, but not all subgroups returned necessarily fulfill the condition.</p>

<p><a id="X833C51BD7E7812C4" name="X833C51BD7E7812C4"></a></p>

<h5>39.21-5 ExactSizeConsiderFunction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ExactSizeConsiderFunction</code>( <var class="Arg">size</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a function <code class="code">consider</code> of four arguments that can be used in <code class="func">SubgroupsSolvableGroup</code> (<a href="chap39.html#X7AD7804A803910AC"><span class="RefLink">39.21-3</span></a>) for the option <code class="code">consider</code> to compute subgroups whose sizes are exactly <var class="Arg">size</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">l:=SubgroupsSolvableGroup(g,rec(actions:=[hom],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">consider:=ExactSizeConsiderFunction(6)));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(l,Size);</span>
[ 1, 3, 9, 27, 54, 6, 108, 216 ]
</pre></div>

<p>Again, the <code class="code">consider</code> function does not provide a perfect filter. It is guaranteed that all subgroups fulfilling the condition are returned, but not all subgroups returned necessarily fulfill the condition.</p>

<p><a id="X7A2C774B7CFF3E07" name="X7A2C774B7CFF3E07"></a></p>

<h5>39.21-6 InfoPcSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InfoPcSubgroup</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>Information function for the subgroup lattice functions using pcgs.</p>

<p><a id="X79F894537D526B61" name="X79F894537D526B61"></a></p>

<h4>39.22 <span class="Heading">Special Generating Sets</span></h4>

<p><a id="X82FD78AF7F80A0E2" name="X82FD78AF7F80A0E2"></a></p>

<h5>39.22-1 GeneratorsSmallest</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; GeneratorsSmallest</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a "smallest" generating set for the group <var class="Arg">G</var>. This is the lexicographically (using <strong class="pkg">GAP</strong>s order of group elements) smallest list <span class="SimpleMath">l</span> of elements of <var class="Arg">G</var> such that <span class="SimpleMath">G = ⟨ l ⟩</span> and <span class="SimpleMath">l_i not ∈ ⟨ l_1, ..., l_{i-1} ⟩</span> (in particular <span class="SimpleMath">l_1</span> is not the identity element of the group). The comparison of two groups via lexicographic comparison of their sorted element lists yields the same relation as lexicographic comparison of their smallest generating sets.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">GeneratorsSmallest(g);</span>
[ (3,4), (2,3), (1,2) ]
</pre></div>

<p><a id="X7A258CCF79552198" name="X7A258CCF79552198"></a></p>

<h5>39.22-2 LargestElementGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LargestElementGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the largest element of <var class="Arg">G</var> with respect to the ordering <code class="code">&lt;</code> of the elements family.</p>

<p><a id="X81D15723804771E2" name="X81D15723804771E2"></a></p>

<h5>39.22-3 MinimalGeneratingSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MinimalGeneratingSet</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a generating set of <var class="Arg">G</var> of minimal possible length.</p>

<p>Note that –apart from special cases– currently there are only efficient methods known to compute minimal generating sets of finite solvable groups and of finitely generated nilpotent groups. Hence so far these are the only cases for which methods are available. The former case is covered by a method implemented in the <strong class="pkg">GAP</strong> library, while the second case requires the package <strong class="pkg">Polycyclic</strong>.</p>

<p>If you do not really need a minimal generating set, but are satisfied with getting a reasonably small set of generators, you better use <code class="func">SmallGeneratingSet</code> (<a href="chap39.html#X814DBABC878D5232"><span class="RefLink">39.22-4</span></a>).</p>

<p>Information about the minimal generating sets of the finite simple groups of order less than <span class="SimpleMath">10^6</span> can be found in <a href="chapBib.html#biBMY79">[MY79]</a>. See also the package <strong class="pkg">AtlasRep</strong>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">MinimalGeneratingSet(g);</span>
[ (2,4,3), (1,4,2,3) ]
</pre></div>

<p><a id="X814DBABC878D5232" name="X814DBABC878D5232"></a></p>

<h5>39.22-4 SmallGeneratingSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SmallGeneratingSet</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a generating set of <var class="Arg">G</var> which has few elements. As neither irredundancy, nor minimal length is proven it runs much faster than <code class="func">MinimalGeneratingSet</code> (<a href="chap39.html#X81D15723804771E2"><span class="RefLink">39.22-3</span></a>). It can be used whenever a short generating set is desired which not necessarily needs to be optimal.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SmallGeneratingSet(g);</span>
[ (1,2,3,4), (1,2) ]
</pre></div>

<p><a id="X7D1574457B152333" name="X7D1574457B152333"></a></p>

<h5>39.22-5 IndependentGeneratorsOfAbelianGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IndependentGeneratorsOfAbelianGroup</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of generators <span class="SimpleMath">a_1, a_2, ...</span> of prime power order or infinite order of the abelian group <var class="Arg">A</var> such that <var class="Arg">A</var> is the direct product of the cyclic groups generated by the <span class="SimpleMath">a_i</span>. The list of orders of the returned generators must match the result of <code class="func">AbelianInvariants</code> (<a href="chap39.html#X812827937F403300"><span class="RefLink">39.16-1</span></a>) (taking into account that zero and <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>) are identified).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=AbelianGroup(IsPermGroup,[15,14,22,78]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(IndependentGeneratorsOfAbelianGroup(g),Order);</span>
[ 2, 2, 2, 3, 3, 5, 7, 11, 13 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariants(g);</span>
[ 2, 2, 2, 3, 3, 5, 7, 11, 13 ]
</pre></div>

<p><a id="X86F835DA8264A0CE" name="X86F835DA8264A0CE"></a></p>

<h5>39.22-6 IndependentGeneratorExponents</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IndependentGeneratorExponents</code>( <var class="Arg">G</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For an abelian group <var class="Arg">G</var>, with <code class="func">IndependentGeneratorsOfAbelianGroup</code> (<a href="chap39.html#X7D1574457B152333"><span class="RefLink">39.22-5</span></a>) value the list <span class="SimpleMath">[ a_1, ..., a_n ]</span>, this operation returns the exponent vector <span class="SimpleMath">[ e_1, ..., e_n ]</span> to represent <span class="SimpleMath"><var class="Arg">g</var> = ∏_i a_i^{e_i}</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g := AbelianGroup([16,9,625]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">gens := IndependentGeneratorsOfAbelianGroup(g);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(gens, Order);</span>
[ 9, 16, 625 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariants(g);</span>
[ 9, 16, 625 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">r:=gens[1]^4*gens[2]^12*gens[3]^128;;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IndependentGeneratorExponents(g,r);</span>
[ 4, 12, 128 ]
</pre></div>

<p><a id="X7CA0B6A27E0BE6B8" name="X7CA0B6A27E0BE6B8"></a></p>

<h4>39.23 <span class="Heading">1-Cohomology</span></h4>

<p>Let <span class="SimpleMath">G</span> be a finite group and <span class="SimpleMath">M</span> an elementary abelian normal <span class="SimpleMath">p</span>-subgroup of <span class="SimpleMath">G</span>. Then the group of 1-cocycles <span class="SimpleMath">Z^1( G/M, M )</span> is defined as</p>

<p class="pcenter">Z^1(G/M, M) = { γ: G/M → M ∣ ∀ g_1, g_2 ∈ G : γ(g_1 M ⋅ g_2 M ) = γ(g_1 M)^{g_2} ⋅ γ(g_2 M) }</p>

<p>and is a <span class="SimpleMath">GF(p)</span>-vector space.</p>

<p>The group of 1-coboundaries <span class="SimpleMath">B^1( G/M, M )</span> is defined as</p>

<p class="pcenter">B^1(G/M, M) = { γ : G/M → M ∣ ∃ m ∈ M ∀ g ∈ G : γ(gM) = (m^{-1})^g ⋅ m }</p>

<p>It also is a <span class="SimpleMath">GF(p)</span>-vector space.</p>

<p>Let <span class="SimpleMath">α</span> be the isomorphism of <span class="SimpleMath">M</span> into a row vector space <span class="SimpleMath">cal W</span> and <span class="SimpleMath">(g_1, ..., g_l)</span> representatives for a generating set of <span class="SimpleMath">G/M</span>. Then there exists a monomorphism <span class="SimpleMath">β</span> of <span class="SimpleMath">Z^1( G/M, M )</span> in the <span class="SimpleMath">l</span>-fold direct sum of <span class="SimpleMath">cal W</span>, such that <span class="SimpleMath">β( γ ) = ( α( γ(g_1 M) ),..., α( γ(g_l M) ) )</span> for every <span class="SimpleMath">γ ∈ Z^1( G/M, M )</span>.</p>

<p><a id="X847BEC137A49BAF4" name="X847BEC137A49BAF4"></a></p>

<h5>39.23-1 <span class="Heading">OneCocycles</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OneCocycles</code>( <var class="Arg">G</var>, <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OneCocycles</code>( <var class="Arg">G</var>, <var class="Arg">mpcgs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OneCocycles</code>( <var class="Arg">gens</var>, <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OneCocycles</code>( <var class="Arg">gens</var>, <var class="Arg">mpcgs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Computes the group of 1-cocycles <span class="SimpleMath">Z^1(<var class="Arg">G</var>/<var class="Arg">M</var>,<var class="Arg">M</var>)</span>. The normal subgroup <var class="Arg">M</var> may be given by a (Modulo)Pcgs <var class="Arg">mpcgs</var>. In this case the whole calculation is performed modulo the normal subgroup defined by <code class="code">DenominatorOfModuloPcgs(<var class="Arg">mpcgs</var>)</code> (see <a href="chap45.html#X7F18A01785DBAC4E"><span class="RefLink">45.1</span></a>). Similarly the group <var class="Arg">G</var> may instead be specified by a set of elements <var class="Arg">gens</var> that are representatives for a generating system for the factor group <var class="Arg">G</var>/<var class="Arg">M</var>. If this is done the 1-cocycles are computed with respect to these generators (otherwise the routines try to select suitable generators themselves). The current version of the code assumes that <var class="Arg">G</var> is a permutation group or a pc group.</p>

<p><a id="X7E6438D5834ACCDA" name="X7E6438D5834ACCDA"></a></p>

<h5>39.23-2 OneCoboundaries</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OneCoboundaries</code>( <var class="Arg">G</var>, <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the group of 1-coboundaries. Syntax of input and output otherwise is the same as with <code class="func">OneCocycles</code> (<a href="chap39.html#X847BEC137A49BAF4"><span class="RefLink">39.23-1</span></a>) except that entries that refer to cocycles are not computed.</p>

<p>The operations <code class="func">OneCocycles</code> (<a href="chap39.html#X847BEC137A49BAF4"><span class="RefLink">39.23-1</span></a>) and <code class="func">OneCoboundaries</code> return a record with (at least) the components:</p>


<dl>
<dt><strong class="Mark"><code class="code">generators</code></strong></dt>
<dd><p>Is a list of representatives for a generating set of <var class="Arg">G</var>/<var class="Arg">M</var>. Cocycles are represented with respect to these generators.</p>

</dd>
<dt><strong class="Mark"><code class="code">oneCocycles</code></strong></dt>
<dd><p>A space of row vectors over GF(<span class="SimpleMath">p</span>), representing <span class="SimpleMath">Z^1</span>. The vectors are represented in dimension <span class="SimpleMath">a ⋅ b</span> where <span class="SimpleMath">a</span> is the length of <code class="code">generators</code> and <span class="SimpleMath">p^b</span> the size of <var class="Arg">M</var>.</p>

</dd>
<dt><strong class="Mark"><code class="code">oneCoboundaries</code></strong></dt>
<dd><p>A space of row vectors that represents <span class="SimpleMath">B^1</span>.</p>

</dd>
<dt><strong class="Mark"><code class="code">cocycleToList</code></strong></dt>
<dd><p>is a function to convert a cocycle (a row vector in <code class="code">oneCocycles</code>) to a corresponding list of elements of <var class="Arg">M</var>.</p>

</dd>
<dt><strong class="Mark"><code class="code">listToCocycle</code></strong></dt>
<dd><p>is a function to convert a list of elements of <var class="Arg">M</var> to a cocycle.</p>

</dd>
<dt><strong class="Mark"><code class="code">isSplitExtension</code></strong></dt>
<dd><p>indicates whether <var class="Arg">G</var> splits over <var class="Arg">M</var>. The following components are only bound if the extension splits. Note that if <var class="Arg">M</var> is given by a modulo pcgs all subgroups are given as subgroups of <var class="Arg">G</var> by generators corresponding to <code class="code">generators</code> and thus may not contain the denominator of the modulo pcgs. In this case taking the closure with this denominator will give the full preimage of the complement in the factor group.</p>

</dd>
<dt><strong class="Mark"><code class="code">complement</code></strong></dt>
<dd><p>One complement to <var class="Arg">M</var> in <var class="Arg">G</var>.</p>

</dd>
<dt><strong class="Mark"><code class="code">cocycleToComplement( cyc )</code></strong></dt>
<dd><p>is a function that takes a cocycle from <code class="code">oneCocycles</code> and returns the corresponding complement to <var class="Arg">M</var> in <var class="Arg">G</var> (with respect to the fixed complement <code class="code">complement</code>).</p>

</dd>
<dt><strong class="Mark"><code class="code">complementToCocycle(<var class="Arg">U</var>)</code></strong></dt>
<dd><p>is a function that takes a complement and returns the corresponding cocycle.</p>

</dd>
</dl>
<p>If the factor <var class="Arg">G</var>/<var class="Arg">M</var> is given by a (modulo) pcgs <var class="Arg">gens</var> then special methods are used that compute a presentation for the factor implicitly from the pcgs.</p>

<p>Note that the groups of 1-cocycles and 1-coboundaries are not groups in the sense of <code class="func">Group</code> (<a href="chap39.html#X7D8E473384DE9CD4"><span class="RefLink">39.2-1</span></a>) for <strong class="pkg">GAP</strong> but vector spaces.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">n:=Group((1,2)(3,4),(1,3)(2,4));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">oc:=OneCocycles(g,n);</span>
rec( cocycleToComplement := function( c ) ... end, 
  cocycleToList := function( c ) ... end, 
  complement := Group([ (3,4), (2,4,3) ]), 
  complementGens := [ (3,4), (2,4,3) ], 
  complementToCocycle := function( K ) ... end, 
  factorGens := [ (3,4), (2,4,3) ], generators := [ (3,4), (2,4,3) ], 
  isSplitExtension := true, listToCocycle := function( L ) ... end, 
  oneCoboundaries := &lt;vector space over GF(2), with 2 generators&gt;, 
  oneCocycles := &lt;vector space over GF(2), with 2 generators&gt; )
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">oc.cocycleToList([ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ]);</span>
[ (1,2)(3,4), (1,2)(3,4) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">oc.listToCocycle([(),(1,3)(2,4)]) = Z(2) * [ 0, 0, 1, 0];</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">oc.cocycleToComplement([ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ]);</span>
Group([ (3,4), (1,3,4) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">oc.complementToCocycle(Group((1,2,4),(1,4))) = Z(2) * [ 0, 1, 1, 1 ];</span>
true
</pre></div>

<p>The factor group <span class="SimpleMath">H^1(<var class="Arg">G</var>/<var class="Arg">M</var>, <var class="Arg">M</var>) = Z^1(<var class="Arg">G</var>/<var class="Arg">M</var>, <var class="Arg">M</var>) / B^1(<var class="Arg">G</var>/<var class="Arg">M</var>, <var class="Arg">M</var>)</span> is called the first cohomology group. Currently there is no function which explicitly computes this group. The easiest way to represent it is as a vector space complement to <span class="SimpleMath">B^1</span> in <span class="SimpleMath">Z^1</span>.</p>

<p>If the only purpose of the calculation of <span class="SimpleMath">H^1</span> is the determination of complements it might be desirable to stop calculations once it is known that the extension cannot split. This can be achieved via the more technical function <code class="func">OCOneCocycles</code> (<a href="chap39.html#X80400ABD7F40FAA0"><span class="RefLink">39.23-3</span></a>).</p>

<p><a id="X80400ABD7F40FAA0" name="X80400ABD7F40FAA0"></a></p>

<h5>39.23-3 OCOneCocycles</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OCOneCocycles</code>( <var class="Arg">ocr</var>, <var class="Arg">onlySplit</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the more technical function to compute 1-cocycles. It takes an record <var class="Arg">ocr</var> as first argument which must contain at least the components <code class="code">group</code> for the group and <code class="code">modulePcgs</code> for a (modulo) pcgs of the module. This record will also be returned with components as described under <code class="func">OneCocycles</code> (<a href="chap39.html#X847BEC137A49BAF4"><span class="RefLink">39.23-1</span></a>) (with the exception of <code class="code">isSplitExtension</code> which is indicated by the existence of a <code class="code">complement</code>) but components such as <code class="code">oneCoboundaries</code> will only be computed if not already present.</p>

<p>If <var class="Arg">onlySplit</var> is <code class="keyw">true</code>, <code class="func">OCOneCocycles</code> returns <code class="keyw">false</code> as soon as possible if the extension does not split.</p>

<p><a id="X811E1CF07DABE924" name="X811E1CF07DABE924"></a></p>

<h5>39.23-4 ComplementClassesRepresentativesEA</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ComplementClassesRepresentativesEA</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes complement classes to an elementary abelian normal subgroup <var class="Arg">N</var> via 1-Cohomology. Normally, a user program should call <code class="func">ComplementClassesRepresentatives</code> (<a href="chap39.html#X811B8A4683DDE1F9"><span class="RefLink">39.11-6</span></a>) instead, which also works for a solvable (not necessarily elementary abelian) <var class="Arg">N</var>.</p>

<p><a id="X8199B1D27D487897" name="X8199B1D27D487897"></a></p>

<h5>39.23-5 InfoCoh</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InfoCoh</code></td><td class="tdright">( info class )</td></tr></table></div>
<p>The info class for the cohomology calculations is <code class="func">InfoCoh</code>.</p>

<p><a id="X80A4B0F282977074" name="X80A4B0F282977074"></a></p>

<h4>39.24 <span class="Heading">Schur Covers and Multipliers</span></h4>

<p>Additional attributes and properties of a group can be derived from computing its Schur cover. For example, if <span class="SimpleMath">G</span> is a finitely presented group, the derived subgroup of a Schur cover of <span class="SimpleMath">G</span> is invariant and isomorphic to the <code class="func">NonabelianExteriorSquare</code> (<a href="chap39.html#X8739CD4686301A0E"><span class="RefLink">39.24-5</span></a>) value of <span class="SimpleMath">G</span>, see <a href="chapBib.html#biBBJR87">[BJR87]</a>.</p>

<p><a id="X7F619DDA7DD6C43B" name="X7F619DDA7DD6C43B"></a></p>

<h5>39.24-1 EpimorphismSchurCover</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; EpimorphismSchurCover</code>( <var class="Arg">G</var>[, <var class="Arg">pl</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an epimorphism <span class="SimpleMath">epi</span> from a group <span class="SimpleMath">D</span> onto <var class="Arg">G</var>. The group <span class="SimpleMath">D</span> is one (of possibly several) Schur covers of <var class="Arg">G</var>. The group <span class="SimpleMath">D</span> can be obtained as the <code class="func">Source</code> (<a href="chap32.html#X7DE8173F80E07AB1"><span class="RefLink">32.3-8</span></a>) value of <var class="Arg">epi</var>. The kernel of <span class="SimpleMath">epi</span> is the Schur multiplier of <var class="Arg">G</var>. If <var class="Arg">pl</var> is given as a list of primes, only the multiplier part for these primes is realized. At the moment, <span class="SimpleMath">D</span> is represented as a finitely presented group.</p>

<p><a id="X7DD1E37987612042" name="X7DD1E37987612042"></a></p>

<h5>39.24-2 SchurCover</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SchurCover</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns one (of possibly several) Schur covers of the group <var class="Arg">G</var>.</p>

<p>At the moment this cover is represented as a finitely presented group and <code class="func">IsomorphismPermGroup</code> (<a href="chap43.html#X80B7B1C783AA1567"><span class="RefLink">43.3-1</span></a>) would be needed to convert it to a permutation group.</p>

<p>If also the relation to <var class="Arg">G</var> is needed, <code class="func">EpimorphismSchurCover</code> (<a href="chap39.html#X7F619DDA7DD6C43B"><span class="RefLink">39.24-1</span></a>) should be used.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">epi:=EpimorphismSchurCover(g);</span>
[ f1, f2, f3 ] -&gt; [ (3,4), (2,4,3), (1,3)(2,4) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Source(epi));</span>
48
</pre></div>

<p>If the group becomes bigger, Schur Cover calculations might become unfeasible.</p>

<p>There is another operation, <code class="func">AbelianInvariantsMultiplier</code> (<a href="chap39.html#X792BC39D7CEB1D27"><span class="RefLink">39.24-3</span></a>), which only returns the structure of the Schur Multiplier, and which should work for larger groups as well.</p>

<p><a id="X792BC39D7CEB1D27" name="X792BC39D7CEB1D27"></a></p>

<h5>39.24-3 AbelianInvariantsMultiplier</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AbelianInvariantsMultiplier</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of the abelian invariants of the Schur multiplier of <var class="Arg">G</var>.</p>

<p>At the moment, this operation will not give any information about how to extend the multiplier to a Schur Cover.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariantsMultiplier(g);</span>
[ 2 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariantsMultiplier(AlternatingGroup(6));</span>
[ 2, 3 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariantsMultiplier(SL(2,3));</span>
[  ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariantsMultiplier(SL(3,2));</span>
[ 2 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AbelianInvariantsMultiplier(PSU(4,2));</span>
[ 2 ]
</pre></div>

<p>(Note that the last command from the example will take some time.)</p>

<p>The <strong class="pkg">GAP</strong> 4.4.12 manual contained examples for larger groups e.g. <span class="SimpleMath">M_22</span>. However, some issues that may very rarely (and not easily reproducibly) lead to wrong results were discovered in the code capable of handling larger groups, and in <strong class="pkg">GAP</strong> 4.5 it was replaced by a more reliable basic method. To deal with larger groups, one can use the function <code class="func">SchurMultiplier</code> (<span class="RefLink">???</span>) from the <strong class="pkg">cohomolo</strong> package. Also, additional methods for <code class="func">AbelianInvariantsMultiplier</code> are installed in the <strong class="pkg">Polycyclic</strong> package for pcp-groups.</p>

<p><a id="X819E8AEC835F8CD1" name="X819E8AEC835F8CD1"></a></p>

<h5>39.24-4 Epicentre</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Epicentre</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ExteriorCentre</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>There are various ways of describing the epicentre of a group <var class="Arg">G</var>. It is the smallest normal subgroup <span class="SimpleMath">N</span> of <var class="Arg">G</var> such that <span class="SimpleMath"><var class="Arg">G</var>/N</span> is a central quotient of a group. It is also equal to the Exterior Center of <var class="Arg">G</var>, see <a href="chapBib.html#biBEllis98">[Ell98]</a>.</p>

<p><a id="X8739CD4686301A0E" name="X8739CD4686301A0E"></a></p>

<h5>39.24-5 NonabelianExteriorSquare</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NonabelianExteriorSquare</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Computes the nonabelian exterior square <span class="SimpleMath"><var class="Arg">G</var><var class="Arg">G</var></span> of the group <var class="Arg">G</var>, which for a finitely presented group is the derived subgroup of any Schur cover of <var class="Arg">G</var> (see <a href="chapBib.html#biBBJR87">[BJR87]</a>).</p>

<p><a id="X7E1C8CD77CDB9F71" name="X7E1C8CD77CDB9F71"></a></p>

<h5>39.24-6 EpimorphismNonabelianExteriorSquare</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; EpimorphismNonabelianExteriorSquare</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Computes the mapping <span class="SimpleMath"><var class="Arg">G</var><var class="Arg">G</var><var class="Arg">G</var></span>. The kernel of this mapping is equal to the Schur multiplier of <var class="Arg">G</var>.</p>

<p><a id="X7BF8DB3D8300BB3F" name="X7BF8DB3D8300BB3F"></a></p>

<h5>39.24-7 IsCentralFactor</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsCentralFactor</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This function determines if there exists a group <span class="SimpleMath">H</span> such that <var class="Arg">G</var> is isomorphic to the quotient <span class="SimpleMath">H/Z(H)</span>. A group with this property is called in literature <em>capable</em>. A group being capable is equivalent to the epicentre of <var class="Arg">G</var> being trivial, see <a href="chapBib.html#biBBFS79">[BFS79]</a>.</p>

<p><a id="X7F4240CD782B6032" name="X7F4240CD782B6032"></a></p>

<h5>39.24-8 <span class="Heading">Covering groups of symmetric groups</span></h5>

<p>The covering groups of symmetric groups were classified in <a href="chapBib.html#biBSchur1911">[Sch11]</a>; an inductive procedure to construct faithful, irreducible representations of minimal degree over all fields was presented in <a href="chapBib.html#biBMaas2010">[Maa10]</a>. Methods for <code class="func">EpimorphismSchurCover</code> (<a href="chap39.html#X7F619DDA7DD6C43B"><span class="RefLink">39.24-1</span></a>) are provided for natural symmetric groups which use these representations. For alternating groups, the restriction of these representations are provided, but they may not be irreducible. In the case of degree <span class="SimpleMath">6</span> and <span class="SimpleMath">7</span>, they are not the full covering groups and so matrix representations are just stored explicitly for the six-fold covers.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">EpimorphismSchurCover(SymmetricGroup(15));</span>
[ &lt; immutable compressed matrix 64x64 over GF(9) &gt;, 
  &lt; immutable compressed matrix 64x64 over GF(9) &gt; ] -&gt; 
[ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (1,2) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">EpimorphismSchurCover(AlternatingGroup(15));</span>
[ &lt; immutable compressed matrix 64x64 over GF(9) &gt;, 
  &lt; immutable compressed matrix 64x64 over GF(9) &gt; ] -&gt; 
[ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (13,14,15) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SchurCoverOfSymmetricGroup(12);</span>
&lt;matrix group of size 958003200 with 2 generators&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">DoubleCoverOfAlternatingGroup(12);</span>
&lt;matrix group of size 479001600 with 2 generators&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">BasicSpinRepresentationOfSymmetricGroup( 10, 3, -1 );</span>
[ &lt; immutable compressed matrix 16x16 over GF(9) &gt;, 
  &lt; immutable compressed matrix 16x16 over GF(9) &gt;, 
  &lt; immutable compressed matrix 16x16 over GF(9) &gt;, 
  &lt; immutable compressed matrix 16x16 over GF(9) &gt;, 
  &lt; immutable compressed matrix 16x16 over GF(9) &gt;, 
  &lt; immutable compressed matrix 16x16 over GF(9) &gt;, 
  &lt; immutable compressed matrix 16x16 over GF(9) &gt;, 
  &lt; immutable compressed matrix 16x16 over GF(9) &gt;, 
  &lt; immutable compressed matrix 16x16 over GF(9) &gt; ]
</pre></div>

<p><a id="X7DDA6BC1824F78FD" name="X7DDA6BC1824F78FD"></a></p>

<h5>39.24-9 BasicSpinRepresentationOfSymmetricGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; BasicSpinRepresentationOfSymmetricGroup</code>( <var class="Arg">n</var>, <var class="Arg">p</var>, <var class="Arg">sign</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Constructs the image of the Coxeter generators in the basic spin (projective) representation of the symmetric group of degree <var class="Arg">n</var> over a field of characteristic <span class="SimpleMath"><var class="Arg">p</var> ≥ 0</span>. There are two such representations and <var class="Arg">sign</var> controls which is returned: +1 gives a group where the preimage of an adjacent transposition <span class="SimpleMath">(i,i+1)</span> has order 4, -1 gives a group where the preimage of an adjacent transposition <span class="SimpleMath">(i,i+1)</span> has order 2. If no <var class="Arg">sign</var> is specified, +1 is used by default. If no <var class="Arg">p</var> is specified, 3 is used by default. (Note that the convention of which cover is labelled as +1 is inconsistent in the literature.)</p>

<p><a id="X844CFFDE80F6AD15" name="X844CFFDE80F6AD15"></a></p>

<h5>39.24-10 SchurCoverOfSymmetricGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SchurCoverOfSymmetricGroup</code>( <var class="Arg">n</var>, <var class="Arg">p</var>, <var class="Arg">sign</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Constructs a Schur cover of <code class="code">SymmetricGroup(<var class="Arg">n</var>)</code> as a faithful, irreducible matrix group in characteristic <var class="Arg">p</var> (<span class="SimpleMath"><var class="Arg">p</var> ≠ 2</span>). For <span class="SimpleMath"><var class="Arg">n</var> ≥ 4</span>, there are two such covers, and <var class="Arg">sign</var> determines which is returned: +1 gives a group where the preimage of an adjacent transposition <span class="SimpleMath">(i,i+1)</span> has order 4, -1 gives a group where the preimage of an adjacent transposition <span class="SimpleMath">(i,i+1)</span> has order 2. If no <var class="Arg">sign</var> is specified, +1 is used by default. If no <var class="Arg">p</var> is specified, 3 is used by default. (Note that the convention of which cover is labelled as +1 is inconsistent in the literature.) For <span class="SimpleMath"><var class="Arg">n</var> ≤ 3</span>, the symmetric group is its own Schur cover and <var class="Arg">sign</var> is ignored. For <span class="SimpleMath"><var class="Arg">p</var> = 2</span>, there is no faithful, irreducible representation of the Schur cover unless <span class="SimpleMath"><var class="Arg">n</var> = 1</span> or <span class="SimpleMath"><var class="Arg">n</var> = 3</span>, so <code class="keyw">fail</code> is returned if <span class="SimpleMath"><var class="Arg">p</var> = 2</span>. For <span class="SimpleMath"><var class="Arg">p</var> = 3</span>, <span class="SimpleMath"><var class="Arg">n</var> = 3</span>, the representation is indecomposable, but reducible. The field of the matrix group is generally <code class="code">GF(<var class="Arg">p</var>^2)</code> if <span class="SimpleMath"><var class="Arg">p</var> &gt; 0</span>, and an abelian number field if <span class="SimpleMath"><var class="Arg">p</var> = 0</span>.</p>

<p><a id="X7E0F4896795E34FC" name="X7E0F4896795E34FC"></a></p>

<h5>39.24-11 DoubleCoverOfAlternatingGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DoubleCoverOfAlternatingGroup</code>( <var class="Arg">n</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Constructs a double cover of <code class="code">AlternatingGroup(<var class="Arg">n</var>)</code> as a faithful, completely reducible matrix group in characteristic <var class="Arg">p</var> (<span class="SimpleMath">p ≠ 2</span>) for <span class="SimpleMath">n ≥ 4</span>. For <span class="SimpleMath">n ≤ 3</span>, the alternating group is its own Schur cover, and <code class="keyw">fail</code> is returned. For <span class="SimpleMath">p = 2</span>, there is no faithful, completely reducible representation of the double cover, so <code class="keyw">fail</code> is returned. The field of the matrix group is generally <code class="code">GF(p^2)</code> if <span class="SimpleMath">p&gt;0</span>, and an abelian number field if <span class="SimpleMath">p=0</span>. If <var class="Arg">p</var> is omitted, the default is 3.</p>

<p><a id="X865722987E0E19B6" name="X865722987E0E19B6"></a></p>

<h4>39.25 <span class="Heading">Tests for the Availability of Methods</span></h4>

<p>The following filters and operations indicate capabilities of <strong class="pkg">GAP</strong>. They can be used in the method selection or algorithms to check whether it is feasible to compute certain operations for a given group. In general, they return <code class="keyw">true</code> if good algorithms for the given arguments are available in <strong class="pkg">GAP</strong>. An answer <code class="keyw">false</code> indicates that no method for this group may exist, or that the existing methods might run into problems.</p>

<p>Typical examples when this might happen is with finitely presented groups, for which many of the methods cannot be guaranteed to succeed in all situations.</p>

<p>The willingness of <strong class="pkg">GAP</strong> to perform certain operations may change, depending on which further information is known about the arguments. Therefore the filters used are not implemented as properties but as "other filters" (see <a href="chap13.html#X871597447BB998A1"><span class="RefLink">13.7</span></a> and <a href="chap13.html#X7997705185C7E720"><span class="RefLink">13.8</span></a>).</p>

<p><a id="X798F13EA810FB215" name="X798F13EA810FB215"></a></p>

<h5>39.25-1 CanEasilyTestMembership</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanEasilyTestMembership</code>( <var class="Arg">G</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>This filter indicates whether <strong class="pkg">GAP</strong> can test membership of elements in the group <var class="Arg">G</var> (via the operation <code class="func">\in</code> (<a href="chap30.html#X84B7FA8C7C94400F"><span class="RefLink">30.6-1</span></a>)) in reasonable time. It is used by the method selection to decide whether an algorithm that relies on membership tests may be used.</p>

<p><a id="X7C2A89607BDFD920" name="X7C2A89607BDFD920"></a></p>

<h5>39.25-2 CanEasilyComputeWithIndependentGensAbelianGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanEasilyComputeWithIndependentGensAbelianGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>This filter indicates whether <strong class="pkg">GAP</strong> can in reasonable time compute independent abelian generators of the group <var class="Arg">G</var> (via <code class="func">IndependentGeneratorsOfAbelianGroup</code> (<a href="chap39.html#X7D1574457B152333"><span class="RefLink">39.22-5</span></a>)) and then can decompose arbitrary group elements with respect to these generators using <code class="func">IndependentGeneratorExponents</code> (<a href="chap39.html#X86F835DA8264A0CE"><span class="RefLink">39.22-6</span></a>). It is used by the method selection to decide whether an algorithm that relies on these two operations may be used.</p>

<p><a id="X83245C82835D496C" name="X83245C82835D496C"></a></p>

<h5>39.25-3 CanComputeSize</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanComputeSize</code>( <var class="Arg">dom</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This filter indicates whether the size of the domain <var class="Arg">dom</var> (which might be <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>)) can be computed.</p>

<p><a id="X8268965487364912" name="X8268965487364912"></a></p>

<h5>39.25-4 CanComputeSizeAnySubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanComputeSizeAnySubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>This filter indicates whether <strong class="pkg">GAP</strong> can easily compute the size of any subgroup of the group <var class="Arg">G</var>. (This is for example advantageous if one can test that a stabilizer index equals the length of the orbit computed so far to stop early.)</p>

<p><a id="X82DDE00D82A32083" name="X82DDE00D82A32083"></a></p>

<h5>39.25-5 CanComputeIndex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanComputeIndex</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This function indicates whether the index <span class="SimpleMath">[<var class="Arg">G</var>:<var class="Arg">H</var>]</span> (which might be <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>)) can be computed. It assumes that <span class="SimpleMath"><var class="Arg">H</var><var class="Arg">G</var></span> (see <code class="func">CanComputeIsSubset</code> (<a href="chap39.html#X7BE7C36B84C23511"><span class="RefLink">39.25-6</span></a>)).</p>

<p><a id="X7BE7C36B84C23511" name="X7BE7C36B84C23511"></a></p>

<h5>39.25-6 CanComputeIsSubset</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanComputeIsSubset</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This filter indicates that <strong class="pkg">GAP</strong> can test (via <code class="func">IsSubset</code> (<a href="chap30.html#X79CA175481F8105F"><span class="RefLink">30.5-1</span></a>)) whether <var class="Arg">B</var> is a subset of <var class="Arg">A</var>.</p>

<p><a id="X87D62C2C7C375E2D" name="X87D62C2C7C375E2D"></a></p>

<h5>39.25-7 KnowsHowToDecompose</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; KnowsHowToDecompose</code>( <var class="Arg">G</var>[, <var class="Arg">gens</var>] )</td><td class="tdright">( property )</td></tr></table></div>
<p>Tests whether the group <var class="Arg">G</var> can decompose elements in the generators <var class="Arg">gens</var>. If <var class="Arg">gens</var> is not given it tests, whether it can decompose in the generators given in the <code class="func">GeneratorsOfGroup</code> (<a href="chap39.html#X79C44528864044C5"><span class="RefLink">39.2-4</span></a>) value of <var class="Arg">G</var>.</p>

<p>This property can be used for example to check whether a group homomorphism by images (see <code class="func">GroupHomomorphismByImages</code> (<a href="chap40.html#X7F348F497C813BE0"><span class="RefLink">40.1-1</span></a>)) can be reasonably defined from this group.</p>


<div class="chlinkprevnextbot">&nbsp;<a href="chap0.html">[Top of Book]</a>&nbsp;  <a href="chap0.html#contents">[Contents]</a>&nbsp;  &nbsp;<a href="chap38.html">[Previous Chapter]</a>&nbsp;  &nbsp;<a href="chap40.html">[Next Chapter]</a>&nbsp;  </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chap16.html">16</a>  <a href="chap17.html">17</a>  <a href="chap18.html">18</a>  <a href="chap19.html">19</a>  <a href="chap20.html">20</a>  <a href="chap21.html">21</a>  <a href="chap22.html">22</a>  <a href="chap23.html">23</a>  <a href="chap24.html">24</a>  <a href="chap25.html">25</a>  <a href="chap26.html">26</a>  <a href="chap27.html">27</a>  <a href="chap28.html">28</a>  <a href="chap29.html">29</a>  <a href="chap30.html">30</a>  <a href="chap31.html">31</a>  <a href="chap32.html">32</a>  <a href="chap33.html">33</a>  <a href="chap34.html">34</a>  <a href="chap35.html">35</a>  <a href="chap36.html">36</a>  <a href="chap37.html">37</a>  <a href="chap38.html">38</a>  <a href="chap39.html">39</a>  <a href="chap40.html">40</a>  <a href="chap41.html">41</a>  <a href="chap42.html">42</a>  <a href="chap43.html">43</a>  <a href="chap44.html">44</a>  <a href="chap45.html">45</a>  <a href="chap46.html">46</a>  <a href="chap47.html">47</a>  <a href="chap48.html">48</a>  <a href="chap49.html">49</a>  <a href="chap50.html">50</a>  <a href="chap51.html">51</a>  <a href="chap52.html">52</a>  <a href="chap53.html">53</a>  <a href="chap54.html">54</a>  <a href="chap55.html">55</a>  <a href="chap56.html">56</a>  <a href="chap57.html">57</a>  <a href="chap58.html">58</a>  <a href="chap59.html">59</a>  <a href="chap60.html">60</a>  <a href="chap61.html">61</a>  <a href="chap62.html">62</a>  <a href="chap63.html">63</a>  <a href="chap64.html">64</a>  <a href="chap65.html">65</a>  <a href="chap66.html">66</a>  <a href="chap67.html">67</a>  <a href="chap68.html">68</a>  <a href="chap69.html">69</a>  <a href="chap70.html">70</a>  <a href="chap71.html">71</a>  <a href="chap72.html">72</a>  <a href="chap73.html">73</a>  <a href="chap74.html">74</a>  <a href="chap75.html">75</a>  <a href="chap76.html">76</a>  <a href="chap77.html">77</a>  <a href="chap78.html">78</a>  <a href="chap79.html">79</a>  <a href="chap80.html">80</a>  <a href="chap81.html">81</a>  <a href="chap82.html">82</a>  <a href="chap83.html">83</a>  <a href="chap84.html">84</a>  <a href="chap85.html">85</a>  <a href="chap86.html">86</a>  <a href="chap87.html">87</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>