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
##
|
