/usr/share/gap/lib/grppcnrm.gi is in gap-libs 4r6p5-3.
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 | #############################################################################
##
#W grppcnrm.gi GAP Library Frank Celler
##
##
#Y Copyright (C) 1996, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the methods for normalizers of polycylic groups.
##
#############################################################################
##
#F PCGS_STABILIZER( <pcgs>, <pnt>, <op> ) . . . . . . . . . . . . . . local
##
PCGS_STABILIZER := function( arg )
local pcgs, pnt, op, data, one, orb, prod, n, s, i,
mi, np, j, o, len, l1, k, l2, r, e, stab, ros,dict;
pcgs := arg[1];
pnt := arg[2];
op := arg[3];
one := OneOfPcgs(pcgs);
ros := RelativeOrders(pcgs);
pcgs := ShallowCopy(pcgs);
dict:=NewDictionary(pnt,true,true);
# without data blob
if Length(arg) = 3 then
# operate on canonical versions
pnt := op( pnt, one );
# store representatives in <r>
orb := [ pnt ];
AddDictionary(dict,pnt,1);
prod := [ 1 ];
n := [];
s := [];
stab := [];
# go *up* the composition series
for i in Reversed([1..Length(pcgs)]) do
mi := pcgs[i];
np := op( pnt, mi );
# is <np> really a new point or is it in <orb>
j := LookupDictionary(dict, np );
# add it if it is new
if j = fail then
o := ros[i];
Add( prod, prod[Length(prod)] * o );
Add( n, i );
len := Length(orb);
l1 := 0;
for k in [ 1 .. o-1 ] do
l2 := l1 + len;
for j in [ 1 .. len ] do
orb[j+l2] := op( orb[j+l1], mi );
AddDictionary(dict,orb[j+l2],j+l2);
od;
l1 := l2;
od;
# if it is the start point the element stabilizes
elif j = 1 then
Add( s, mi );
# compute a stabilizing element
else
if not IsBound(stab[j]) then
r := one;
l1 := j-1;
len := Length(prod);
for k in [ 1 .. len-1 ] do
e := QuoInt( l1, prod[len-k] );
r := pcgs[n[len-k]]^e * r;
l1 := l1 mod prod[len-k];
if l1 = 0 then
break;
fi;
od;
stab[j] := r;
fi;
Add( s, pcgs[i] / stab[j] );
fi;
od;
# with data blob
else
data := arg[4];
# operate on canonical versions
pnt := op( data, pnt, one );
# store representatives in <r>
orb := [ pnt ];
AddDictionary(dict,pnt,1);
prod := [ 1 ];
n := [];
s := [];
stab := [];
# go *up* the composition series
for i in Reversed([1..Length(pcgs)]) do
mi := pcgs[i];
np := op( data, pnt, mi );
# is <np> really a new point or is it in <orb>
j := LookupDictionary(dict, np );
# add it if it is new
if j = fail then
o := ros[i];
Add( prod, prod[Length(prod)] * o );
Add( n, i );
len := Length(orb);
l1 := 0;
for k in [ 1 .. o-1 ] do
l2 := l1 + len;
for j in [ 1 .. len ] do
orb[j+l2] := op( data, orb[j+l1], mi );
AddDictionary(dict,orb[j+l2],j+l2);
od;
l1 := l2;
od;
# if it is the start point the element stabilizes
elif j = 1 then
Add( s, mi );
# compute a stabilizing element
else
if not IsBound(stab[j]) then
r := one;
l1 := j-1;
len := Length(prod);
for k in [ 1 .. len-1 ] do
e := QuoInt( l1, prod[len-k] );
r := pcgs[n[len-k]]^e * r;
l1 := l1 mod prod[len-k];
if l1 = 0 then
break;
fi;
od;
stab[j] := r;
fi;
Add( s, pcgs[i] / stab[j] );
fi;
od;
fi;
Info( InfoPcNormalizer, 3, "orbit length: ", Length(orb) );
return Reversed(s);
end;
#############################################################################
##
#F PCGS_STABILIZER_HOMOMORPHIC( <pcgs>, <homs>, <pnt>, <op> ) . . . . local
##
PCGS_STABILIZER_HOMOMORPHIC := function( arg )
local pcgs, homs, pnt, op, ros, one, hone, orb, prod,
n, s, stab, i, mi, np, j, o, len, l1, k, l2,
r, e, data,dict;
pcgs := arg[1];
homs := arg[2];
pnt := arg[3];
op := arg[4];
dict:=NewDictionary(pnt,true,true);
if 0 = Length(pcgs) then
return pcgs;
fi;
if Length(pcgs) <> Length(homs) then
Error( "expecting ", Length(pcgs), " homomorphic images in <homs>" );
fi;
ros := RelativeOrders(pcgs);
one := OneOfPcgs(pcgs);
hone := One(homs[1]);
pcgs := ShallowCopy(pcgs);
# without data blob
if Length(arg) = 4 then
# operate on canonical versions
pnt := op( pnt, hone );
# store representatives in <r>
orb := [ pnt ];
AddDictionary(dict,pnt,1);
prod := [ 1 ];
n := [];
s := [];
stab := [];
# go *up* the composition series
for i in Reversed([1..Length(pcgs)]) do
mi := homs[i];
np := op( pnt, mi );
# is <np> really a new point or is it in <orb>
j := LookupDictionary(dict, np );
# add it if it is new
if j = fail then
o := ros[i];
Add( prod, prod[Length(prod)] * o );
Add( n, i );
len := Length(orb);
l1 := 0;
for k in [ 1 .. o-1 ] do
l2 := l1 + len;
for j in [ 1 .. len ] do
orb[j+l2] := op( orb[j+l1], mi );
AddDictionary(dict,orb[j+l2],j+l2);
od;
l1 := l2;
od;
# if it is the start point the element stabilizes
elif j = 1 then
Add( s, pcgs[i] );
# compute a stabilizing element
else
if not IsBound(stab[j]) then
r := one;
l1 := j-1;
len := Length(prod);
for k in [ 1 .. len-1 ] do
e := QuoInt( l1, prod[len-k] );
r := pcgs[n[len-k]]^e * r;
l1 := l1 mod prod[len-k];
if l1 = 0 then
break;
fi;
od;
stab[j] := r;
fi;
Add( s, pcgs[i] / stab[j] );
fi;
od;
# with data blob, this case is not used at all
else
Error("you should never be here");
fi;
Info( InfoPcNormalizer, 3, "orbit length: ", Length(orb) );
return Reversed(s);
end;
#############################################################################
##
#F PCGS_NORMALIZER( <home>, <norm>, <point>, <pcgs>, <modulo> )
##
PCGS_NORMALIZER_OPB := function( home, elm, obj )
local ord;
elm := elm^obj;
ord := RelativeOrderOfPcElement( home, elm );
return elm ^ ( 1 / LeadingExponentOfPcElement( home, elm ) mod ord );
end;
PCGS_NORMALIZER_OPC1 := function( data, elm, obj )
local ord;
elm := elm^obj;
ord := RelativeOrderOfPcElement( data[1], elm );
elm := elm ^ ( 1 / LeadingExponentOfPcElement( data[1], elm ) mod ord );
return HeadPcElementByNumber( data[1], elm, data[2] );
end;
PCGS_NORMALIZER_OPC2 := function( data, elm, obj )
# was: return CanonicalPcElement( data[2], elm^obj );
local ord;
elm := elm^obj;
ord:=RelativeOrderOfPcElement(data[1],elm);
elm := elm ^ ( 1 / LeadingExponentOfPcElement( data[1], elm ) mod ord );
return CanonicalPcElement( data[2], elm );
end;
PCGS_NORMALIZER_OPD := function( data, lst, obj )
lst:=CorrespondingGeneratorsByModuloPcgs(data,List(lst,i->i^obj));
return lst;
end;
PCGS_NORMALIZER_OPE := function( data, lst, obj )
local home, pag, pos, max, i, g, dg, exp, j, ros;
home := data[1];
pag := data[2]; # make sure to reset <pag> before returning
pos := [];
max := data[3];
ros := data[4];
for i in [ Length(lst), Length(lst)-1 .. 1 ] do
g := lst[i]^obj;
dg := DepthOfPcElement( home, g );
while dg < max do
if IsBound(pag[dg]) then
g := ReducedPcElement( home, g, pag[dg] );
dg := DepthOfPcElement( home, g );
else
pag[dg] := g;
AddSet( pos, dg );
break;
fi;
od;
od;
for i in Reversed(pos) do
exp := LeadingExponentOfPcElement( home, pag[i] );
if exp <> 1 then
pag[i] := pag[i] ^ (1/exp mod ros[i]);
fi;
for j in [ i+1 .. max-1 ] do
if IsBound(pag[j]) then
exp := ExponentOfPcElement( home, pag[i], j );
if exp <> 0 then
pag[i] := pag[i] * pag[j]^(ros[j]-exp);
fi;
fi;
od;
pag[i] := HeadPcElementByNumber( home, pag[i], max );
od;
lst := pag{pos};
for i in pos do Unbind(pag[i]); od;
return lst;
end;
PCGS_NORMALIZER_DATAE := function( home, modulo )
local id, ros, sub, i, dg, exp, max;
id := OneOfPcgs(home);
ros := RelativeOrders(home);
sub := [];
for i in modulo do
dg := DepthOfPcElement( home, i );
exp := LeadingExponentOfPcElement( home, i );
if exp <> 1 then
i := i ^ (1/exp mod ros[dg]);
fi;
sub[dg] := i;
od;
max := Length(home)+1;
while 2 <= max and IsBound(sub[max-1]) do
max := max-1;
od;
return [ home, sub, max, ros ];
end;
PCGS_NORMALIZER := function( home, pcgs, pnt, modulo )
local op, s, data;
Info( InfoPcNormalizer, 5, "home: ", ShallowCopy(home) );
Info( InfoPcNormalizer, 4, "normalizer: ", ShallowCopy(pcgs) );
Info( InfoPcNormalizer, 4, "point: ", ShallowCopy(pnt) );
Info( InfoPcNormalizer, 5, "modulo: ", ShallowCopy(modulo) );
# if <pnt> and <modulo> have the same length nothing is to be done
if Length(pnt) = Length(modulo) then
Info( InfoPcNormalizer, 3, "PCGS_NORMALIZER case A" );
return pcgs;
# if <pnt> mod <modulo> has only one element operate on elements
elif Length(pnt)-1 = Length(modulo) then
if 0 = Length(modulo) then
Info( InfoPcNormalizer, 3, "PCGS_NORMALIZER case B" );
pnt := pnt[1];
op := PCGS_NORMALIZER_OPB;
data := home;
s := PCGS_STABILIZER( pcgs, pnt, op, home );
else
pnt := pnt mod modulo;
pnt := pnt[1];
if ParentPcgs(modulo)=home and IsTailInducedPcgsRep(modulo) then
Info( InfoPcNormalizer, 3, "PCGS_NORMALIZER case C1" );
op := PCGS_NORMALIZER_OPC1;
data := [ home, modulo!.tailStart ];
else
Info( InfoPcNormalizer, 3, "PCGS_NORMALIZER case C2" );
op := PCGS_NORMALIZER_OPC2;
data := [home,modulo];
fi;
s := PCGS_STABILIZER( pcgs, pnt, op, data );
fi;
# if the <modulo> is trivial it is relatively easy
elif 0 = Length(modulo) then
Info( InfoPcNormalizer, 3, "PCGS_NORMALIZER case D" );
op := PCGS_NORMALIZER_OPD;
pnt := ShallowCopy(pnt);
s := PCGS_STABILIZER( pcgs, pnt, op, home );
# it is get more complicated
else
Info( InfoPcNormalizer, 3, "PCGS_NORMALIZER case E" );
data := PCGS_NORMALIZER_DATAE( home, modulo );
op := PCGS_NORMALIZER_OPE;
pnt := ShallowCopy( pnt mod modulo );
s := PCGS_STABILIZER( pcgs, pnt, op, data );
fi;
# convert it into a modulo pcgs
pcgs := SumPcgs( home, DenominatorOfModuloPcgs(pcgs), s )
mod DenominatorOfModuloPcgs(pcgs);
Info( InfoPcNormalizer, 4, "new norm: ", ShallowCopy(pcgs) );
return pcgs;
end;
#############################################################################
##
#F PCGS_NORMALIZER_LINEAR( <home>, <norm>, <point>, <modulo-pcgs> )
##
PCGS_NORMALIZER_LINEAR := function( home, pcgs, pnt, modulo )
local f, o, m, sub, s,p,op;
Info( InfoPcNormalizer, 5, "home: ", ShallowCopy(home) );
Info( InfoPcNormalizer, 4, "normalizer: ", ShallowCopy(pcgs) );
Info( InfoPcNormalizer, 4, "point: ", ShallowCopy(pnt) );
Info( InfoPcNormalizer, 5, "modulo: ", ShallowCopy(modulo) );
# construct the linear operation
p:=RelativeOrderOfPcElement( home, modulo[1] );
f := GF(p);
o := One(f);
m := List( pcgs, x -> List( modulo, y ->
ExponentsConjugateLayer( modulo, y,x ) * o ) );
for s in [1..Length(m)] do
m[s]:=ImmutableMatrix(f,m[s]);
od;
# convert <pnt> into a subspace
sub := pnt mod DenominatorOfModuloPcgs(modulo);
sub := List( sub, x -> ExponentsOfPcElement( modulo, x ) * o );
sub:=ImmutableMatrix(f,sub);
# select operation function and prepare matrices if necessary
if p=2 then
op:=OnSubspacesByCanonicalBasisGF2;
else
op:=OnSubspacesByCanonicalBasis;
fi;
# compute the stabilizer
Info( InfoPcNormalizer, 3, "PCGS_NORMALIZER_LINEAR case A" );
s := PCGS_STABILIZER_HOMOMORPHIC( pcgs, m, sub, op );
# convert it into a modulo pcgs
pcgs := SumPcgs( home, DenominatorOfModuloPcgs(pcgs), s )
mod DenominatorOfModuloPcgs(pcgs);
Info( InfoPcNormalizer, 4, "new norm: ", ShallowCopy(pcgs) );
return pcgs;
end;
#############################################################################
##
#F PCGS_CONJUGATING_WORD_GS( <home>, <n>, <u>, <v>, <k> )
##
## Let <u> / <k> and <v> / <k> be two p-groups such that <u>*<n> = <v>*<n>
## and let <n> be an elementary abelian q-group with q <> p. Then a word x
## of <n> with <u> ^ x = <v> is returned. <k> must be normal in <u>*<n>.
##
## It is important, that the weights of <K> are less than those of <N>.
##
PCGS_CONJUGATING_WORD_GS := function( home, n, u, v, k )
local id, x, q, i, p, t, m, vv, mm, xx, j;
# if <n> or <u> / <k> is trivial, just return identity
id := OneOfPcgs(home);
if 0 = Length(n) or 0 = Length(u) or u = v then
return id;
fi;
# Find the word <n> using the algorithm of Kantor. See S.P.Glasby and
# Michael C. Slattery, "Computing intersections and normalizers in
# soluble groups", 1989.
x := id;
q := RelativeOrderOfPcElement( home, n[1] );
for i in Reversed( [ 1 .. Length(u) ] ) do
# the orders must be coprime
p := RelativeOrderOfPcElement( home, u[i] );
if q = p then
Error( "relative orders <u> and <n> are not coprime" );
fi;
# Compute an integer <t> such that <t> * <p> = -1 mod <q>.
t := -Gcdex( p, q ).coeff1;
while t > q do t := t - q; od;
while t < 0 do t := t + q; od;
m := LeftQuotient( u[i]^x, v[i] );
m := SiftedPcElement( k, m );
vv := id;
mm := id;
xx := id;
# construct the product m^v * (m^2)^(v^2) * ... * (m^p-1)^(v^p-1)
for j in [ 1 .. p-1 ] do
vv := vv * v[i];
mm := mm * m;
xx := xx * ( mm^vv );
od;
x := x * ( xx ^ t );
od;
return x;
end;
#############################################################################
##
#F PCGS_NORMALIZER_GLASBY( <home>, <norm>, <nis>, <pcgs>, <modulo> )
##
PCGS_NORMALIZER_GLASBY := function( home, pcgs, nis, u1, u2 )
local id, stb, data, pnt, i, cnj, ns, one, mats, sys,
sol, v, j;
# The situtation is as follows:
#
# S
# \
# \
# Us
# / \
# / \
# U1 Ns N
# \ / \ /
# \ / \ /
# U2 NiS
# \ /
# \ /
# Un
#
# and <S> stabilizes <U2>
# first correct (S mod NiS)
Info( InfoPcNormalizer, 4, "correcting glasby block stabilizer" );
id := OneOfPcgs(pcgs);
stb := NumeratorOfModuloPcgs(pcgs) mod NumeratorOfModuloPcgs(nis);
stb := ShallowCopy(stb);
data := PCGS_NORMALIZER_DATAE( home, u2 );
pnt := PCGS_NORMALIZER_OPE( data, u1 mod u2, id );
for i in [ 1 .. Length(stb) ] do
cnj := PCGS_NORMALIZER_OPE( data, pnt, stb[i] );
cnj := PCGS_CONJUGATING_WORD_GS( home, nis, cnj, pnt, u2 );
stb[i] := stb[i] * cnj;
od;
# now compute the stabilizer in <nis>
Info( InfoPcNormalizer, 4, "computing the centralizer in <nis>" );
# first the operation of <pnt> on (NiS mod U2)
ns := SumPcgs( home, u2, NumeratorOfModuloPcgs(nis) ) mod u2;
one := One( GF(RelativeOrderOfPcElement(home,ns[1])) );
mats := List( pnt, x -> List( ns, y ->
ExponentsConjugateLayer( ns, y,x ) * one ) );
# set up the system of equations
one := One(mats[1]);
sys := [];
for i in [ 1 .. Length(mats[1]) ] do
sys[i] := [];
for j in [ 1 .. Length(mats) ] do
Append( sys[i], one[i] - mats[j][i] );
od;
od;
sol := TriangulizedNullspaceMat(sys);
for v in sol do
v := List( v, IntFFE );
Add( stb, PcElementByExponentsNC(ns,v) );
od;
# Now we have the normalizer in <S> / <U2>. Get the complete preimage.
return SumPcgs( home, u2, stb )
mod DenominatorOfModuloPcgs(pcgs);
end;
#############################################################################
##
#F PCGS_NORMALIZER_COBOUNDS( <home>, <norm>, <nis>, <pcgs>, <modulo> )
##
PCGS_NORMALIZER_COBOUNDS := function( home, pcgs, nis, u1, u2 )
local ns, us, gf, one, data, u, ui, mats, t, l, i, b,
nb, c, heads, k, ln1, ln2, op, stab, s, j, v;
# The situtation is as follows:
#
# S
# \
# \
# Us
# / \
# / \
# U1 Ns N
# \ / \ /
# \ / \ /
# U2 NiS
# \ /
# \ /
# Un
#
# and <S> stabilizes <U2>
# compute the operation of <u1> mod <u2> on <ns> mod <u2>
ns := SumPcgs( home, u2, NumeratorOfModuloPcgs(nis) ) mod u2;
us := SumPcgs( home, u1, NumeratorOfModuloPcgs(nis) );
gf := GF(RelativeOrderOfPcElement(home,ns[1]));
one := One(gf);
data := PCGS_NORMALIZER_DATAE( home, u2 );
u := PCGS_NORMALIZER_OPE( data, u1 mod u2, OneOfPcgs(home) );
ui := List( u, Inverse );
mats := List( u, x -> List(ns, y -> ExponentsConjugateLayer(ns,y,x)*one) );
# compute the coboundaries
Info( InfoPcNormalizer, 4, "using coboundaries and centralizer" );
t := One(mats[1]);
l := [];
for i in [ 1 .. Length(mats[1]) ] do
l[i] := [];
for j in [ 1 .. Length(mats) ] do
Append( l[i], t[i]-mats[j][i] );
od;
od;
b := TriangulizedGeneratorsByMatrix( ns, l, gf );
nb := b[1];
b := b[2];
for i in b do
ConvertToVectorRep(i,gf);
od;
# trivial coboundaries, use ordinary orbit
if IsEmpty(b) then
Info( InfoPcNormalizer, 4, "coboundaries are trivial" );
return PCGS_NORMALIZER( home, pcgs, u1, u2 );
fi;
Info( InfoPcNormalizer, 4, "|coboundaries| = ",
RelativeOrderOfPcElement(home,ns[1]), "^", Length(b) );
# compute the stabilizer
c := List( TriangulizedNullspaceMat(l), x -> PcElementByExponentsNC(ns,x) );
# compute the heads of the coboundaries
heads := [];
k := 1;
i := 1;
while i <= Length(b) and k <= Length(b[1]) do
if IntFFE(b[i][k]) <> 0 then
heads[i] := k;
i := i+1;
fi;
k := k+1;
od;
# now the function which acts on the coboundaries
ln1 := Length(ns);
ln2 := Length(u);
op := function( v, x )
local w, i;
# add the coboundary <v> to <u>
w := ShallowCopy(u);
for i in [ 1 .. ln2 ] do
w[i] := w[i] * PcElementByExponentsNC(ns, v{[(i-1)*ln1+1..i*ln1]});
od;
# operate with <x> on <w> and normalize modulo <u2>
w := PCGS_NORMALIZER_OPE( data, w, x );
# convert back into a vector
v := [];
for i in [ 1 .. ln2 ] do
Append( v, ExponentsOfPcElement( ns, ui[i]*w[i] ) );
od;
v := v * One(gf);
ConvertToVectorRep(v,gf);
for i in [ 1 .. Length(heads) ] do
v := v - v[heads[i]] * b[i];
od;
return Immutable(v);
end;
# compute the blockstabilizer
Info( InfoPcNormalizer, 4, "computing blockstabilizer" );
stab := PCGS_STABILIZER( NumeratorOfModuloPcgs(pcgs) mod us,
b[1] * Zero(gf),
op );
# compute and correct the blockstabilizer
Info( InfoPcNormalizer, 4, "correcting blockstabilizer" );
nb := List( nb, x -> x ^ -1 );
for i in [ 1 .. Length(stab) ] do
s := PCGS_NORMALIZER_OPE( data, u, stab[i] );
v := [];
for j in [ 1 .. ln2 ] do
Append( v, ExponentsOfPcElement( ns, ui[j]*s[j] ) );
od;
for j in [ 1 .. Length(heads) ] do
if v[heads[j] ] <> 0 then
stab[i] := stab[i] * ( nb[j]^v[heads[j]] );
fi;
od;
od;
# return sum of <L>, <C> and <U1>
return InducedPcgsByGeneratorsNC( home, Concatenation( stab, c, u1 ) )
mod DenominatorOfModuloPcgs(pcgs);
end;
#############################################################################
##
#F PcGroup_NormalizerWrtHomePcgs( <u>, <f1>, <f2>, <f3>, <f4> )
##
## compute the normalizer of <u> in its home pcgs, the flags <f1> to <f4>
## can be used to fine tune the normalizer computation:
##
## <f1> if 'true', intersections with the same prime than the module are
## computed using one cobounds. Otherwise an ordinary orbit
## stabilizer algorithm is used.
##
## <f2> if 'true', intersections with different prime than the module are
## computed using one cobounds. Otherwise the method of computation
## depends on the flag <f3>.
##
## <f3> if 'true' and <f2> is 'false', then intersections with different
## prime than the module are computed using Glasby's algorithm.
## Otherwise a ordinary orbit stabilizer algorithm is used.
##
## <f4> if 'true', the first intersection is computed using linear
## operations. Otherwise a ordinary orbit stabilizer algorithm is
## used.
##
PcGroup_NormalizerWrtHomePcgs := function( u, f1, f2, f3, f4 )
local g, # home pcgs of <pcgs>
e, r, # elementary abelian series of <G> and its length
ue, # factor pcgs <pcgs><e>[i] mod <e>[i]
uk, uj, ui_1, # intersections of <pcgs> with <e>[x]
s, si_1, # stabilizer and its intersection with <e>[i-1]
ei_1, # <e>[i-1] mod <e>[i]
pj, pi_1, # primes of <e>[j] and <e>[i-1]
st, # used for checking the algorithm
i, j, k, # loops
pcgs, # pcgs of <u>
id, # identity element
tmp; # temporary
# get the parent pcgs and the elementary abelian series
g := HomePcgs(u);
id := OneOfPcgs(g);
e := ElementaryAbelianSubseries(g);
if e = fail then
Info( InfoPcNormalizer, 1, "Computing el.ab. PCGS" );
s := SpecialPcgs(g);
k := NaturalIsomorphismByPcgs( GroupOfPcgs(g), s );
if ElementaryAbelianSubseries(Pcgs(Image(k))) = fail then
Error( "corrupted special pcgs" );
fi;
tmp := InducedPcgsByGeneratorsNC( g, List(
PcGroup_NormalizerWrtHomePcgs( Image(k,u), f1, f2, f3, f4 ),
x -> PreImage( k, x ) ) );
SetHomePcgs( tmp, g );
return tmp;
fi;
r := Length(e);
# get a canonical pcgs for <u>
pcgs := CanonicalPcgsWrtHomePcgs(u);
# If <r> = 2, <g> is abelian, so we can return <g>
if r = 2 then
return g;
fi;
# compute the closure of <pcgs> and <e>[i]
ue := [];
for i in [ 1 .. r ] do
ue[i] := SumPcgs( g, e[i], pcgs );
od;
# begin with <g>/<e>[2], in this factorgroup nothing is to be done
s := e[1] mod e[2];
Info( InfoPcNormalizer, 1, "skiping level 1 of ", r );
Info( InfoPcNormalizer, 1, "skiping level 2 of ", r );
# start with <g>/<e>[3] because <g>/<e>[2] is abelian
for i in [ 3 .. r ] do
# <s> = Normalizer( <G>/<E>[i-1], <pcgs> )
#
# The first step looks like ( U = <pcgs> )
#
# S
# \
# \
# U Ei-1
# \ /
# \ /
# Ui-1
# \
# \
# Ei
#
# Now get the complete preimage of <s> in <g>/<e>[i] and start the
# whole computation for that factorgroup.
s := NumeratorOfModuloPcgs(s) mod e[i];
Info( InfoPcNormalizer, 1, "reached level ", i, " of ", r );
Info( InfoPcNormalizer, 4, "normalizer: ", AsList(s) );
Info( InfoPcNormalizer, 4, "subgroup: ", AsList(ue[i]) );
Info( InfoPcNormalizer, 5, "modulo: ", AsList(e[i]) );
# keep the old stabilizer for an assert later
st := s;
# if <ue>[i] is trivial we can skip this step
ei_1 := e[i-1] mod e[i];
if Length(ue[i]) = Length(e[i]) then
Info( InfoPcNormalizer, 2, "<ue>[", i, "] is trivial" );
Assert( 1, IsNormal(GroupOfPcgs(st),GroupOfPcgs(ue[i])) );
# if <e>[i-1] is a subgroup of <ue>[i] we can skip this step
elif ForAll( ei_1, x -> SiftedPcElement(ue[i],x) = id ) then
Info( InfoPcNormalizer, 2, "<e>[",i,"] > <ue>[",i-1,"]" );
Assert( 1, IsNormal(GroupOfPcgs(st),GroupOfPcgs(ue[i])) );
# now do some real work
else
# remember the prime of the current section for later
pi_1 := RelativeOrderOfPcElement( g, ei_1[1] );
# get the first section
ui_1 := NormalIntersectionPcgs( g, e[i-1], ue[i] );
# if the factor is trivial do nothing
if Length(ui_1) = Length(e[i]) then
Info( InfoPcNormalizer, 2,
"<ue>[",i,"] /\\ <e>[",i-1,"] is trivial" );
# if <f4> is true, use linear operations
elif f4 then
Info( InfoPcNormalizer, 2, "<ue>[", i, "] /\\ <e>[", i-1,
"] using linear operation" );
s := PCGS_NORMALIZER_LINEAR( g, s, ui_1, ei_1 );
# otherwise use a normal stabilizer
else
Info( InfoPcNormalizer, 2, "<ue>[", i, "] /\\ <e>[", i-1,
"] using orbit" );
s := PCGS_NORMALIZER( g, s, ui_1, e[i] );
fi;
# check the stabilizer
Assert( 3, Stabilizer( GroupOfPcgs(st), GroupOfPcgs(ui_1),
function(U,g) return U^g;end)
= GroupOfPcgs(s) );
# now <ui_1> must be stabilized by <s>
st := s;
Assert( 1, IsNormal(GroupOfPcgs(st),GroupOfPcgs(ui_1)) );
# find <ue>[i]/\<E>[j] which is larger then <ue>[i]/\<E>[i-1]
j := i-2;
uj := NormalIntersectionPcgs( g, e[j], ue[i] );
k := i-1;
uk := ui_1;
while 0 < j and Length(uj) = Length(ui_1) do
Info( InfoPcNormalizer, 2, "<ue>[",i,"] /\\ <e>[", j,
"] = <ue>[", i, "] /\\ e[", k, "]" );
k := j;
uk := uj;
j := j - 1;
if 0 < j then
uj := NormalIntersectionPcgs( g, e[j], ue[i] );
fi;
od;
# The next step for <s> = Normalizer( <uk> ) is
#
# S
# \ Ej
# \ / \
# U ** \
# \ / \ Ek
# \ / \ / \
# Uj ** \
# \ / \ Ei-1
# \ / \ /
# Uk Si-1
# \ /
# \ /
# Ui-1
# \
# \
# Ei
#
# If <j> = 0 or <s> and <u> have the same <E>[i-1] intersection
# we are finished with this step.
si_1 := NormalIntersectionPcgs(
g,
e[i-1],
NumeratorOfModuloPcgs(s) )
mod e[i];
while 0<j and not ForAll(si_1,x ->SiftedPcElement(ui_1,x)=id) do
# this only works for subseries <e>
tmp := First( e[j], x -> not x in e[j+1] );
pj := RelativeOrderOfPcElement( g, tmp );
# cobounds
if ( pj = pi_1 and f1 ) or ( pj <> pi_1 and f2 ) then
Info( InfoPcNormalizer, 2, "<ue>[", i, "] /\\ <e>[", j,
"] using cobounds" );
s := PCGS_NORMALIZER_COBOUNDS( g, s, si_1, uj, uk );
# glasby
elif pj <> pi_1 and f3 then
Info( InfoPcNormalizer, 2, "<ue>[", i, "] /\\ <e>[", j,
"] using Glasby" );
s := PCGS_NORMALIZER_GLASBY( g, s, si_1, uj, uk );
# orbit
else
Info( InfoPcNormalizer, 2, "<ue>[", i, "] /\\ <e>[", j,
"] using orbit" );
s := PCGS_NORMALIZER( g, s, uj, uk );
fi;
# check the stabilizer
Assert( 3, Stabilizer( GroupOfPcgs(st), GroupOfPcgs(uj),
function(U,g) return U^g;end)
= GroupOfPcgs(s) );
# now <uj> must be stabilized by <s>
st := s;
Assert( 1, IsNormal(GroupOfPcgs(st),GroupOfPcgs(uj)) );
# find the next non-trivial intersection
k := j;
uk := uj;
while 0 < j and Length(uj) = Length(uk) do
if k <> j then
Info( InfoPcNormalizer, 2, "<ue>[", i, "] /\\ <e>[",
j, "] = <ue>[", i, "] /\\ e[", k, "]" );
fi;
k := j;
uk := uj;
j := j - 1;
if 0 < j then
uj := NormalIntersectionPcgs( g, e[j], ue[i] );
fi;
od;
# Now we know our new <S>, if <j>-1 is still nonzero, compute
# the intersection in order to see, if we are finshed.
if 0 < j then
si_1 := NormalIntersectionPcgs(
g,
e[i-1],
NumeratorOfModuloPcgs(s) )
mod e[i];
fi;
od;
fi;
od;
Assert( 1, IsNormal( GroupOfPcgs(s), u ) );
if Length(s) = Length(pcgs) then
return pcgs;
else
tmp := InducedPcgsByPcSequence( g, List( s, x -> x ) );
SetHomePcgs( tmp, g );
return tmp;
fi;
end;
#############################################################################
##
#M NormalizerInHomePcgs( <pc-group> )
##
InstallMethod( NormalizerInHomePcgs,
"for group with home pcgs",
true,
[ IsGroup and HasHomePcgs ],
0,
function( u )
if not IsPrimeOrdersPcgs(HomePcgs(u)) then
TryNextMethod();
fi;
return PcGroup_NormalizerWrtHomePcgs( u, true, false, true, true );
end );
#############################################################################
##
#M Normalizer( <pc-group>, <pc-group> )
##
InstallMethod( NormalizerOp, "for groups with home pcgs", IsIdenticalObj,
[ IsGroup and HasHomePcgs, IsGroup and HasHomePcgs ],
1, #better than the next method
function( g, u )
local home, norm, pcgs;
# for small groups use direct calculation
if Size(g) < 1000 or (Size(g)<100000 and Size(g)/Size(u)<500) then
TryNextMethod();
fi;
home := HomePcgs(g);
if home <> HomePcgs(u) then
TryNextMethod();
fi;
# first compute the normalizer with respect to the home
pcgs := NormalizerInHomePcgs(u);
norm := SubgroupByPcgs( g, pcgs );
# then the intersection
norm := Intersection( g, norm );
# and return
return norm;
end );
InstallMethod( NormalizerOp, "slightly better orbit algorithm for pc groups",
IsIdenticalObj, [ IsGroup and HasHomePcgs, IsGroup and HasHomePcgs ], 0,
function( G, U )
local N,h,opfun;
h:=HomePcgs(G);
opfun:=function(p,g)
return CanonicalPcgs(InducedPcgsByGeneratorsNC(h,List(p,i->i^g)));
end;
N:=Stabilizer(G,CanonicalPcgs(InducedPcgs(h,U)),opfun);
return N;
end);
#############################################################################
##
#E grppcnrm.gi . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
##
|