This file is indexed.

/usr/share/gap/lib/randiso2.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
#############################################################################
##
#W  randiso2.gi               GAP library                  Hans Ulrich Besche
##
#Y  Copyright (C)  1997,  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
##

#############################################################################
##
#F  EvalFpCoc( coc, desc ). . . . . . . . . . . . . . . . . . . . . . . local
##
EvalFpCoc := function( coc, desc )
    local powers, exp, targets, result, i, j, g1, g2, fcd4, pos, map;

    if desc[ 1 ] = 1 then
        # test, if g^i in cl(g)
        return List( coc[ desc[ 2 ] ],
                     function( x )
                     if x[ 1 ] ^ desc[ 3 ] in x then return 1; fi; return 0;
                     end );

    elif desc[ 1 ] = 2 then
        # test, if cl(g) is root of cl(h)
        exp := QuoInt( Order( coc[ desc[ 2 ] ][ 1 ][ 1 ] ),
                       Order( coc[ desc[ 3 ] ][ 1 ][ 1 ] ) );
        powers := Flat( coc[ desc[ 3 ] ] );
        return List( coc[ desc[ 2 ] ],
                     function(x)
                     if x[ 1 ] ^ exp in powers then return 1; fi; return 0;
                     end );

    elif desc[ 1 ] = 3 then
        # test, if cl(g) is power of cl(h)
        exp := QuoInt( Order( coc[ desc[ 3 ] ][ 1 ][ 1 ] ),
                       Order( coc[ desc[ 2 ] ][ 1 ][ 1 ] ) );
        # just one representative for each class of power-candidates
        powers := List( coc[ desc[ 2 ] ], x -> x[ 1 ] );
        result := List( powers, x -> 0 );
        for i in List( Flat( coc[ desc[ 3 ] ] ), x -> x ^ exp ) do
            for j in [ 1 .. Length( powers ) ] do
                if i = powers[ j ] then
                    result[ j ] := result[ j ] + 1;
                fi;
            od;
        od;
        return result;

    else 
        # test how often the word [ a, b ] * a^2 is hit
        targets := List( coc[ desc[ 2 ] ], x -> x[ 1 ] );
        map := [ 1 .. Length( targets ) ];
        SortParallel( targets, map );
        result := List( targets, x -> 0 );
        fcd4 := Flat( coc[ desc[ 4 ] ] );
        for g1 in Flat( coc[ desc[ 3 ] ] ) do
            for g2 in fcd4 do
                if desc[ 1 ] = 4 then 
                    pos := Position( targets, Comm( g1, g2 ) * g1 ^ 2 );
                else 
                # desc[ 1 ] = 5
                    pos := Position( targets, Comm( g1, g2 ) * g1 ^ 3 );
                fi;
                if not IsBool( pos ) then
                    result[ map[ pos ] ] := result[ map[ pos ] ] + 1;
                fi;
            od;
        od;
        return result;
    fi;
end;

#############################################################################
##
#F CocGroup( G ). . . . . . . . . . . . . . . . . . . . . . . . . . . . local
##
CocGroup := function( g )

   local orbs, typs, styps, coc, i, j;

   # compute the conjugacy classes of G as lists of elements and
   # classify them according to representative order and length
   orbs  := OrbitsDomain( g, AsList( g ) );
   typs  := List( orbs, x -> [ Order( x[ 1 ] ), Length( x ) ] );
   styps := Set( typs );
   coc   := List( styps, x-> [ ] );
   for i in [ 1 .. Length( styps ) ] do
      for j in [ 1 .. Length( orbs ) ] do
         if styps[ i ] = typs[ j ] then
            Add( coc[ i ], orbs[ j ] );
         fi;
      od;
   od;
   return coc;
end;

#############################################################################
##
#F DiffCoc( coc, pos, finps ) . . . . . . . . . . . . . . . . . . . . . local
##
DiffCoc := function( coc, pos, finps )

   local tmp, sfinps, i, j;

   # split up the pos-th cluster of coc using the fingerprint-values finps
   sfinps := Set( finps );
   tmp := List( sfinps, x -> [ ] );
   for i in [ 1 .. Length( sfinps ) ] do
      for j in [ 1 .. Length( finps ) ] do
         if sfinps[ i ] = finps[ j ] then
            Add( tmp[ i ], coc[ pos ][ j ] );
         fi;
      od;
   od;
   return Concatenation( coc{[1..pos-1]}, tmp, coc{[pos+1..Length(coc)]} );
   end;

#############################################################################
##
#F SplitUpSublistsByFpFunc( list ). . . . . . . . . . . . . . . . . . . local
##
SplitUpSublistsByFpFunc := function( list )

   local result, finp, finps, i, g, j;

   result := [ ];
   finps := [ ];
   for i in [ 1 .. Length( list ) ] do
      if list[ i ].isUnique then 
         Add( result, [ list [ i ] ] );
         Add( finps, false );
      else
         g    := PcGroupCodeRec( list[i] );
         finp := FingerprintFF( g );
         j    := Position( finps, finp );
         if IsBool( j ) then
            Add( result, [ list[ i ] ] );
            Add( finps, finp );
            Info( InfoRandIso, 3, "split into ", Length( finps ),
                  " classes within ", i, " of ", Length( list ), " tests" );
         else
            Add( result[ j ], list[ i ] );
            if i mod 50 = 0 then
              Info( InfoRandIso, 3, "still ", Length( finps ),
                    " classes after ", i, " of ", Length( list ), " tests" );
            fi;
         fi;
      fi;
   od;
   for i in [ 1 .. Length( result ) ] do
      if Length( result[ i ] ) = 1 then
         result[ i ] := result[ i ][ 1 ];
         result[ i ].isUnique := true;
      fi;
   od;
   Info( InfoRandIso, 2, "   Iso: found ", Length(result)," classes incl. ",
          Length( Filtered( result, IsRecord ) )," unique groups");
   return result;
end;

#############################################################################
##
#F CodeGenerators( gens, spcgs ). . . . . . . . . . . . . . . . . . . . local
##
CodeGenerators := function( gens, spcgs )

   local  layers, first, one, pcgs, sgrps, dep, lay, 
          numf, pos, e, tpos, found, et, p;

   gens   := ShallowCopy( gens );
   layers := LGLayers( spcgs );
   first  := LGFirst( spcgs );
   one    := OneOfPcgs( spcgs );
   pcgs   := [ ];
   sgrps  := [ ];
   
   numf   := 0;
   pos    := 0;

   while numf < Length( spcgs ) do
      pos := pos + 1;
      e   := gens[ pos ];
      while e <> one do

         dep := DepthOfPcElement( spcgs, e );
         lay := layers[ dep ];
         tpos := first[ lay + 1 ];
         found := false;
         
         while tpos > first[ lay ] and not found and e <> one do
            tpos := tpos - 1;
            if not IsBound( pcgs[ tpos ] ) then
               pcgs[ tpos ] := e;
               sgrps[ tpos ] := GroupByGenerators( Concatenation( [ e ],
                                pcgs{[ tpos + 1 .. first[ lay + 1 ] - 1 ]},
                                spcgs{[ first[lay+1] .. Length(spcgs) ]} ) );
               for p in Set( FactorsInt( Order( e ) ) ) do
                  et := e ^ p;
                  if et <> one and not et in gens then
                     Add( gens, et );
                  fi;
               od;
               for p in Compacted( pcgs ) do
                  et := Comm( e, p );
                  if et <> one and not et in gens then
                     Add( gens, et );
                  fi;
               od;
               e := one;
               numf := numf + 1;
            else
               if e in sgrps[ tpos ] then
                  found := true;
               fi;
            fi;
         od;
         if found then
            while tpos < first[ lay + 1 ] do
               if tpos + 1 = first[ lay + 1 ] then
                  while e <> one and
                        lay = layers[ DepthOfPcElement( spcgs, e ) ] do
                     e := pcgs[ tpos ] ^ -1 * e;
                  od;
               else
                  while not e in sgrps[ tpos + 1 ] do
                     e := pcgs[ tpos ] ^ -1 * e;
                  od;
               fi;
               tpos := tpos + 1;
            od;
         fi;
      od;
   od;
   pcgs := PcgsByPcSequenceNC( ElementsFamily( FamilyObj( spcgs ) ), pcgs );
   SetRelativeOrders( pcgs, RelativeOrders( spcgs ) );
   return rec( pcgs := pcgs, code := CodePcgs( pcgs ) );
end;

#############################################################################
##
#F IsomorphismSolvableSmallGroups( G, H  ). . . . . isomorphism from G onto H
##
IsomorphismSolvableSmallGroups := function( g, h )
   local size, coc1, coc2, lcoc, coclen, p, poses, nposes, i, qual, nqual,
         lmin, spcgs1, spcgs2, gens, code, gens1, gens2, codes1, codes2,
         G, H, iso, iso1, iso2;

   size := Size( g );
   if size <> Size( h ) then
      return fail;
   fi;
   if size = 1 then
     return GroupHomomorphismByImagesNC( g, h, [], [] );
   fi;
   if ID_AVAILABLE( size ) = fail or size > 2000 then
      Error( "IsomorphismSmallSolvableGroups: groups are not small" );
   fi;
   if IdGroup( g ) <> IdGroup( h ) then
      return fail;
   fi;
   if not IsSolvableGroup( g ) then
      Error( "IsomorphismSmallSolvableGroups: groups are not solvable" );
   fi;

   if IsPcGroup( g ) then
      G := g;
   else
      iso1 := IsomorphismPcGroup( g );
      G := Image( iso1 );
   fi;
   if IsPcGroup( h ) then
      H := h;
   else
      iso2 := IsomorphismPcGroup( h );
      H := Image( iso2 );
   fi;

   coc1 := CocGroup( G );
   coc1 := List( coc1{[ 2 .. Length( coc1 ) ]}, Concatenation );
   coc2 := CocGroup( H );
   coc2 := List( coc2{[ 2 .. Length( coc2 ) ]}, Concatenation );
   lcoc := Length( coc1 );
   coclen := List( coc1, Length );

   lmin := Length( MinimalGeneratingSet( G ) );
   qual := size ^ lmin;
   poses := fail;
   i := - Length( FactorsInt( size ) ) * 5 - lcoc * 8 - lmin * 12;
   Info( InfoRandIso, 3, "testing ", -i, " generating strategies" );
   while poses = fail or i < 0 do
      i := i + 1;
      nposes := List( [ 1 .. lmin ], x -> Random( [ 1 .. lcoc ] ) );
      nqual := Product( coclen{ nposes } );
      if nqual < qual and
          Size( Group( List( coc1{ nposes }, Random ) ) ) = size then
         qual := nqual;
         poses := nposes;
      fi;
   od;
   Info( InfoRandIso, 2, "strategy with ",qual," generating set candidates");

   coc1 := coc1{ poses };
   coc2 := coc2{ poses };
   gens1 := [];
   gens2 := [];
   codes1 := [];
   codes2 := [];
   spcgs1 := SpecialPcgs( G );
   spcgs2 := SpecialPcgs( H );
   iso := fail;
   i := 0;

   while iso = fail do
      i := i + 1;
      if i mod 10 = 0 then
         Info( InfoRandIso, 3, i, " test on generating set candidates" );
      fi;
      if gens1 = [] then
         gens := ShallowCopy( GeneratorsOfGroup( G ) );
      else
         gens := List( coc1, Random );
      fi;
      if Size( Group( gens ) ) = size then
         code := CodeGenerators( gens, spcgs1 );
         p := Position( codes2, code.code );
         if p <> fail then
            iso := GroupHomomorphismByImagesNC( G, H, code.pcgs,
                                 CodeGenerators( gens2[ p ], spcgs2 ).pcgs );
         fi;
         if not code.code in codes1 then
            Add( codes1, code.code );
            Add( gens1, gens );
         fi;
      fi;
      if iso = fail then
         if gens2 = [] then
            gens := ShallowCopy( GeneratorsOfGroup( H ) );
         else
            gens := List( coc2, Random );
         fi;
         if Size( Group( gens ) ) = size then
            code := CodeGenerators( gens, spcgs2 );
            p := Position( codes1, code.code );
            if p <> fail then
               iso := GroupHomomorphismByImagesNC( G, H,
                       CodeGenerators( gens1[ p ], spcgs1 ).pcgs, code.pcgs);
            fi;
            if not code.code in codes2 then
               Add( codes2, code.code );
               Add( gens2, gens );
            fi;
         fi;
      fi;
   od;

   gens := GeneratorsOfGroup( g );
   if IsBound( iso1 ) then
      gens := List( gens, x -> Image( iso1, x ) );
   fi;
   gens := List( gens, x -> Image( iso, x ) );
   if IsBound( iso2 ) then
      gens := List( gens, x -> PreImage( iso2, x ) );
   fi;
   return GroupHomomorphismByImagesNC( g, h, GeneratorsOfGroup( g ), gens );
end;