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
|
<?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 (polycyclic) - Chapter 7: Higher level methods for pcp-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="chap7" 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="chapA.html">A</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap6.html">[Previous Chapter]</a> <a href="chap8.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap7_mj.html">[MathJax on]</a></p>
<p><a id="X85BB6FE078679DAF" name="X85BB6FE078679DAF"></a></p>
<div class="ChapSects"><a href="chap7.html#X85BB6FE078679DAF">7 <span class="Heading">Higher level methods for pcp-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X8266A0A2821D98A1">7.1 <span class="Heading">Subgroup series in pcp-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X8037DAD77A19D9B2">7.1-1 PcpSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X86C633357ACD342C">7.1-2 EfaSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X80ED4F8380DC477E">7.1-3 SemiSimpleEfaSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7A879948834BD889">7.1-4 DerivedSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X866D4C5C79F26611">7.1-5 RefinedDerivedSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X86F7DE927DE3B5CD">7.1-6 RefinedDerivedSeriesDown</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X879D55A67DB42676">7.1-7 LowerCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X8428592E8773CD7B">7.1-8 UpperCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X83CA5DE785AE3F2C">7.1-9 TorsionByPolyEFSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7E39431286969377">7.1-10 PcpsBySeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X79789A1C82139854">7.1-11 PcpsOfEfaSeries</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7CE2DA437FD2B383">7.2 <span class="Heading">Orbit stabilizer methods for pcp-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X83E17DB483B33AB5">7.2-1 PcpOrbitStabilizer</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X80694BA480F69A0E">7.2-2 StabilizerIntegralAction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X875BE4077B32A411">7.2-3 NormalizerIntegralAction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X80E3B42E792532B3">7.3 <span class="Heading">Centralizers, Normalizers and Intersections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X808EE8AD7EE3ECE1">7.3-1 Centralizer</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X849B5C527BAFAAA4">7.3-2 Centralizer</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X851069107CACF98E">7.3-3 Intersection</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7CF015E87A2B2388">7.4 <span class="Heading">Finite subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X8036FA507A170DC4">7.4-1 TorsionSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X8082CD337972DC63">7.4-2 NormalTorsionSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X86D92DA17DCE22DD">7.4-3 IsTorsionFree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X819058217B4F3DC0">7.4-4 FiniteSubgroupClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7E7C32EA81A297B6">7.4-5 FiniteSubgroupClassesBySeries</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7D9F737F80F6E396">7.5 <span class="Heading">Subgroups of finite index and maximal subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X87D62D497A8715FB">7.5-1 MaximalSubgroupClassesByIndex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7800133F81BC7674">7.5-2 LowIndexSubgroupClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7F7067C77F2DC32C">7.5-3 LowIndexNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X85A5BC447D83175F">7.5-4 NilpotentByAbelianNormalSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X785E0E877AB1D549">7.6 <span class="Heading">Further attributes for pcp-groups based on the Fitting subgroup</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X780552B57C30DD8F">7.6-1 FittingSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X86BD63DC844731DF">7.6-2 IsNilpotentByFinite</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X847ABE6F781C7FE8">7.6-3 Centre</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X861C36368435EB09">7.6-4 FCCentre</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7E75E2BC806746AC">7.6-5 PolyZNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X86800BF783E30D4A">7.6-6 NilpotentByAbelianByFiniteSeries</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X878DBDC77CCA4F7E">7.7 <span class="Heading">Functions for nilpotent groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X81D15723804771E2">7.7-1 MinimalGeneratingSet</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X8640F9D47A1F7434">7.8 <span class="Heading">Random methods for pcp-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X80AEE73E7D639699">7.8-1 RandomCentralizerPcpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X824142B784453DB9">7.9 <span class="Heading">Non-abelian tensor product and Schur extensions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X79EF28D9845878C9">7.9-1 SchurExtension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X84B60EC978A9A05E">7.9-2 SchurExtensionEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7DD1E37987612042">7.9-3 SchurCover</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X792BC39D7CEB1D27">7.9-4 AbelianInvariantsMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X822ED5978647C93B">7.9-5 NonAbelianExteriorSquareEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X8739CD4686301A0E">7.9-6 NonAbelianExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X86553D7B7DABF38F">7.9-7 NonAbelianTensorSquareEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7C0DF7C97F78C666">7.9-8 NonAbelianTensorSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7AE75EC1860FFE7A">7.9-9 NonAbelianExteriorSquarePlusEmbedding</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7D96C84E87925B0F">7.9-10 NonAbelianTensorSquarePlusEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X8746533787C4E8BC">7.9-11 NonAbelianTensorSquarePlus</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X78F9184078B2761A">7.9-12 WhiteheadQuadraticFunctor</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7D3023697BA5CE5A">7.10 <span class="Heading">Schur covers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7D90B44E7B96AFF1">7.10-1 SchurCovers</a></span>
</div></div>
</div>
<h3>7 <span class="Heading">Higher level methods for pcp-groups</span></h3>
<p>This is a description of some higher level functions of the <strong class="pkg">Polycyclic</strong> package of GAP 4. Throughout this chapter we let <var class="Arg">G</var> be a pc-presented group and we consider algorithms for subgroups <var class="Arg">U</var> and <var class="Arg">V</var> of <var class="Arg">G</var>. For background and a description of the underlying algorithms we refer to <a href="chapBib.html#biBEic01b">[Eic01a]</a>.</p>
<p><a id="X8266A0A2821D98A1" name="X8266A0A2821D98A1"></a></p>
<h4>7.1 <span class="Heading">Subgroup series in pcp-groups</span></h4>
<p>Many algorithm for pcp-groups work by induction using some series through the group. In this section we provide a number of useful series for pcp-groups. An <em>efa series</em> is a normal series with elementary or free abelian factors. See <a href="chapBib.html#biBEic00">[Eic00]</a> for outlines on the algorithms of a number of the available series.</p>
<p><a id="X8037DAD77A19D9B2" name="X8037DAD77A19D9B2"></a></p>
<h5>7.1-1 PcpSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PcpSeries</code>( <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the polycyclic series of <var class="Arg">U</var> defined by an igs of <var class="Arg">U</var>.</p>
<p><a id="X86C633357ACD342C" name="X86C633357ACD342C"></a></p>
<h5>7.1-2 EfaSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EfaSeries</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a normal series of <var class="Arg">U</var> with elementary or free abelian factors.</p>
<p><a id="X80ED4F8380DC477E" name="X80ED4F8380DC477E"></a></p>
<h5>7.1-3 SemiSimpleEfaSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemiSimpleEfaSeries</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an efa series of <var class="Arg">U</var> such that every factor in the series is semisimple as a module for <var class="Arg">U</var> over a finite field or over the rationals.</p>
<p><a id="X7A879948834BD889" name="X7A879948834BD889"></a></p>
<h5>7.1-4 DerivedSeriesOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivedSeriesOfGroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>the derived series of <var class="Arg">U</var>.</p>
<p><a id="X866D4C5C79F26611" name="X866D4C5C79F26611"></a></p>
<h5>7.1-5 RefinedDerivedSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RefinedDerivedSeries</code>( <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>the derived series of <var class="Arg">U</var> refined to an efa series such that in each abelian factor of the derived series the free abelian factor is at the top.</p>
<p><a id="X86F7DE927DE3B5CD" name="X86F7DE927DE3B5CD"></a></p>
<h5>7.1-6 RefinedDerivedSeriesDown</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RefinedDerivedSeriesDown</code>( <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>the derived series of <var class="Arg">U</var> refined to an efa series such that in each abelian factor of the derived series the free abelian factor is at the bottom.</p>
<p><a id="X879D55A67DB42676" name="X879D55A67DB42676"></a></p>
<h5>7.1-7 LowerCentralSeriesOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowerCentralSeriesOfGroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>the lower central series of <var class="Arg">U</var>. If <var class="Arg">U</var> does not have a largest nilpotent quotient group, then this function may not terminate.</p>
<p><a id="X8428592E8773CD7B" name="X8428592E8773CD7B"></a></p>
<h5>7.1-8 UpperCentralSeriesOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UpperCentralSeriesOfGroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>the upper central series of <var class="Arg">U</var>. This function always terminates, but it may terminate at a proper subgroup of <var class="Arg">U</var>.</p>
<p><a id="X83CA5DE785AE3F2C" name="X83CA5DE785AE3F2C"></a></p>
<h5>7.1-9 TorsionByPolyEFSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TorsionByPolyEFSeries</code>( <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an efa series of <var class="Arg">U</var> such that all torsion-free factors are at the top and all finite factors are at the bottom. Such a series might not exist for <var class="Arg">U</var> and in this case the function returns fail.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ExamplesOfSomePcpGroups(5);</span>
Pcp-group with orders [ 2, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Igs(G);</span>
[ g1, g2, g3, g4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PcpSeries(G);</span>
[ Pcp-group with orders [ 2, 0, 0, 0 ],
Pcp-group with orders [ 0, 0, 0 ],
Pcp-group with orders [ 0, 0 ],
Pcp-group with orders [ 0 ],
Pcp-group with orders [ ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( PcpSeries(G), Igs );</span>
[ [ g1, g2, g3, g4 ], [ g2, g3, g4 ], [ g3, g4 ], [ g4 ], [ ] ]
</pre></div>
<p>Algorithms for pcp-groups often use an efa series of <span class="SimpleMath">G</span> and work down over the factors of this series. Usually, pcp's of the factors are more useful than the actual factors. Hence we provide the following.</p>
<p><a id="X7E39431286969377" name="X7E39431286969377"></a></p>
<h5>7.1-10 PcpsBySeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PcpsBySeries</code>( <var class="Arg">ser</var>[, <var class="Arg">flag</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list of pcp's corresponding to the factors of the series. If the parameter <var class="Arg">flag</var> is present and equals the string <q>snf</q>, then each pcp corresponds to a decomposition of the abelian groups into direct factors.</p>
<p><a id="X79789A1C82139854" name="X79789A1C82139854"></a></p>
<h5>7.1-11 PcpsOfEfaSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PcpsOfEfaSeries</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of pcps corresponding to an efa series of <var class="Arg">U</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ExamplesOfSomePcpGroups(5);</span>
Pcp-group with orders [ 2, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PcpsBySeries( DerivedSeriesOfGroup(G));</span>
[ Pcp [ g1, g2, g3, g4 ] with orders [ 2, 2, 2, 2 ],
Pcp [ g2^-2, g3^-2, g4^2 ] with orders [ 0, 0, 4 ],
Pcp [ g4^8 ] with orders [ 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PcpsBySeries( RefinedDerivedSeries(G));</span>
[ Pcp [ g1, g2, g3 ] with orders [ 2, 2, 2 ],
Pcp [ g4 ] with orders [ 2 ],
Pcp [ g2^2, g3^2 ] with orders [ 0, 0 ],
Pcp [ g4^2 ] with orders [ 2 ],
Pcp [ g4^4 ] with orders [ 2 ],
Pcp [ g4^8 ] with orders [ 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PcpsBySeries( DerivedSeriesOfGroup(G), "snf" );</span>
[ Pcp [ g2, g3, g1 ] with orders [ 2, 2, 4 ],
Pcp [ g4^2, g3^-2, g2^2*g4^2 ] with orders [ 4, 0, 0 ],
Pcp [ g4^8 ] with orders [ 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">G.1^4 in DerivedSubgroup( G );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">G.1^2 = G.4;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput"> PcpsOfEfaSeries( G );</span>
[ Pcp [ g1 ] with orders [ 2 ],
Pcp [ g2 ] with orders [ 0 ],
Pcp [ g3 ] with orders [ 0 ],
Pcp [ g4 ] with orders [ 0 ] ]
</pre></div>
<p><a id="X7CE2DA437FD2B383" name="X7CE2DA437FD2B383"></a></p>
<h4>7.2 <span class="Heading">Orbit stabilizer methods for pcp-groups</span></h4>
<p>Let <var class="Arg">U</var> be a pcp-group which acts on a set <span class="SimpleMath">Ω</span>. One of the fundamental problems in algorithmic group theory is the determination of orbits and stabilizers of points in <span class="SimpleMath">Ω</span> under the action of <var class="Arg">U</var>. We distinguish two cases: the case that all considered orbits are finite and the case that there are infinite orbits. In the latter case, an orbit cannot be listed and a description of the orbit and its corresponding stabilizer is much harder to obtain.</p>
<p>If the considered orbits are finite, then the following two functions can be applied to compute the considered orbits and their corresponding stabilizers.</p>
<p><a id="X83E17DB483B33AB5" name="X83E17DB483B33AB5"></a></p>
<h5>7.2-1 PcpOrbitStabilizer</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PcpOrbitStabilizer</code>( <var class="Arg">point</var>, <var class="Arg">gens</var>, <var class="Arg">acts</var>, <var class="Arg">oper</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">‣ PcpOrbitsStabilizers</code>( <var class="Arg">points</var>, <var class="Arg">gens</var>, <var class="Arg">acts</var>, <var class="Arg">oper</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The input <var class="Arg">gens</var> can be an igs or a pcp of a pcp-group <var class="Arg">U</var>. The elements in the list <var class="Arg">gens</var> act as the elements in the list <var class="Arg">acts</var> via the function <var class="Arg">oper</var> on the given points; that is, <var class="Arg">oper( point, acts[i] )</var> applies the <span class="SimpleMath">i</span>th generator to a given point. Thus the group defined by <var class="Arg">acts</var> must be a homomorphic image of the group defined by <var class="Arg">gens</var>. The first function returns a record containing the orbit as component 'orbit' and and igs for the stabilizer as component 'stab'. The second function returns a list of records, each record contains 'repr' and 'stab'. Both of these functions run forever on infinite orbits.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralPcpGroup( 0 );</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">mats := [ [[-1,0],[0,1]], [[1,1],[0,1]] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pcp := Pcp(G);</span>
Pcp [ g1, g2 ] with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PcpOrbitStabilizer( [0,1], pcp, mats, OnRight );</span>
rec( orbit := [ [ 0, 1 ] ],
stab := [ g1, g2 ],
word := [ [ [ 1, 1 ] ], [ [ 2, 1 ] ] ] )
</pre></div>
<p>If the considered orbits are infinite, then it may not always be possible to determine a description of the orbits and their stabilizers. However, as shown in <a href="chapBib.html#biBEOs01">[EO02]</a> and <a href="chapBib.html#biBEic02">[Eic02]</a>, it is possible to determine stabilizers and check if two elements are contained in the same orbit if the given action of the polycyclic group is a unimodular linear action on a vector space. The following functions are available for this case.</p>
<p><a id="X80694BA480F69A0E" name="X80694BA480F69A0E"></a></p>
<h5>7.2-2 StabilizerIntegralAction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StabilizerIntegralAction</code>( <var class="Arg">U</var>, <var class="Arg">mats</var>, <var class="Arg">v</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">‣ OrbitIntegralAction</code>( <var class="Arg">U</var>, <var class="Arg">mats</var>, <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first function computes the stabilizer in <var class="Arg">U</var> of the vector <var class="Arg">v</var> where the pcp group <var class="Arg">U</var> acts via <var class="Arg">mats</var> on an integral space and <var class="Arg">v</var> and <var class="Arg">w</var> are elements in this integral space. The second function checks whether <var class="Arg">v</var> and <var class="Arg">w</var> are in the same orbit and the function returns either <var class="Arg">false</var> or a record containing an element in <var class="Arg">U</var> mapping <var class="Arg">v</var> to <var class="Arg">w</var> and the stabilizer of <var class="Arg">v</var>.</p>
<p><a id="X875BE4077B32A411" name="X875BE4077B32A411"></a></p>
<h5>7.2-3 NormalizerIntegralAction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalizerIntegralAction</code>( <var class="Arg">U</var>, <var class="Arg">mats</var>, <var class="Arg">B</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">‣ ConjugacyIntegralAction</code>( <var class="Arg">U</var>, <var class="Arg">mats</var>, <var class="Arg">B</var>, <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first function computes the normalizer in <var class="Arg">U</var> of the lattice with the basis <var class="Arg">B</var>, where the pcp group <var class="Arg">U</var> acts via <var class="Arg">mats</var> on an integral space and <var class="Arg">B</var> is a subspace of this integral space. The second functions checks whether the two lattices with the bases <var class="Arg">B</var> and <var class="Arg">C</var> are contained in the same orbit under <var class="Arg">U</var>. The function returns either <var class="Arg">false</var> or a record with an element in <var class="Arg">U</var> mapping <var class="Arg">B</var> to <var class="Arg">C</var> and the stabilizer of <var class="Arg">B</var>.</p>
<div class="example"><pre>
# get a pcp group and a free abelian normal subgroup
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ExamplesOfSomePcpGroups(8);</span>
Pcp-group with orders [ 0, 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">efa := EfaSeries(G);</span>
[ Pcp-group with orders [ 0, 0, 0, 0, 0 ],
Pcp-group with orders [ 0, 0, 0, 0 ],
Pcp-group with orders [ 0, 0, 0 ],
Pcp-group with orders [ ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">N := efa[3];</span>
Pcp-group with orders [ 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFreeAbelian(N);</span>
true
# create conjugation action on N
<span class="GAPprompt">gap></span> <span class="GAPinput">mats := LinearActionOnPcp(Igs(G), Pcp(N));</span>
[ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
[ [ 0, 0, 1 ], [ 1, -1, 1 ], [ 0, 1, 0 ] ],
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ]
# take an arbitrary vector and compute its stabilizer
<span class="GAPprompt">gap></span> <span class="GAPinput">StabilizerIntegralAction(G,mats, [2,3,4]);</span>
Pcp-group with orders [ 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Igs(last);</span>
[ g1, g3, g4, g5 ]
# check orbits with some other vectors
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitIntegralAction(G,mats, [2,3,4],[3,1,5]);</span>
rec( stab := Pcp-group with orders [ 0, 0, 0, 0 ], prei := g2 )
<span class="GAPprompt">gap></span> <span class="GAPinput">OrbitIntegralAction(G,mats, [2,3,4], [4,6,8]);</span>
false
# compute the orbit of a subgroup of Z^3 under the action of G
<span class="GAPprompt">gap></span> <span class="GAPinput">NormalizerIntegralAction(G, mats, [[1,0,0],[0,1,0]]);</span>
Pcp-group with orders [ 0, 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Igs(last);</span>
[ g1, g2^2, g3, g4, g5 ]
</pre></div>
<p><a id="X80E3B42E792532B3" name="X80E3B42E792532B3"></a></p>
<h4>7.3 <span class="Heading">Centralizers, Normalizers and Intersections</span></h4>
<p>In this section we list a number of operations for which there are methods installed to compute the corresponding features in polycyclic groups.</p>
<p><a id="X808EE8AD7EE3ECE1" name="X808EE8AD7EE3ECE1"></a></p>
<h5>7.3-1 Centralizer</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Centralizer</code>( <var class="Arg">U</var>, <var class="Arg">g</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsConjugate</code>( <var class="Arg">U</var>, <var class="Arg">g</var>, <var class="Arg">h</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>These functions solve the conjugacy problem for elements in pcp-groups and they can be used to compute centralizers. The first method returns a subgroup of the given group <var class="Arg">U</var>, the second method either returns a conjugating element or false if no such element exists.</p>
<p>The methods are based on the orbit stabilizer algorithms described in <a href="chapBib.html#biBEOs01">[EO02]</a>. For nilpotent groups, an algorithm to solve the conjugacy problem for elements is described in <a href="chapBib.html#biBSims94">[Sim94]</a>.</p>
<p><a id="X849B5C527BAFAAA4" name="X849B5C527BAFAAA4"></a></p>
<h5>7.3-2 Centralizer</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Centralizer</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Normalizer</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsConjugate</code>( <var class="Arg">U</var>, <var class="Arg">V</var>, <var class="Arg">W</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>These three functions solve the conjugacy problem for subgroups and compute centralizers and normalizers of subgroups. The first two functions return subgroups of the input group <var class="Arg">U</var>, the third function returns a conjugating element or false if no such element exists.</p>
<p>The methods are based on the orbit stabilizer algorithms described in <a href="chapBib.html#biBEic02">[Eic02]</a>. For nilpotent groups, an algorithm to solve the conjugacy problems for subgroups is described in <a href="chapBib.html#biBLo98">[Lo98b]</a>.</p>
<p><a id="X851069107CACF98E" name="X851069107CACF98E"></a></p>
<h5>7.3-3 Intersection</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Intersection</code>( <var class="Arg">U</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>A general method to compute intersections of subgroups of a pcp-group is described in <a href="chapBib.html#biBEic01b">[Eic01a]</a>, but it is not yet implemented here. However, intersections of subgroups <span class="SimpleMath">U, N ≤ G</span> can be computed if <span class="SimpleMath">N</span> is normalising <span class="SimpleMath">U</span>. See <a href="chapBib.html#biBSims94">[Sim94]</a> for an outline of the algorithm.</p>
<p><a id="X7CF015E87A2B2388" name="X7CF015E87A2B2388"></a></p>
<h4>7.4 <span class="Heading">Finite subgroups</span></h4>
<p>There are various finite subgroups of interest in polycyclic groups. See <a href="chapBib.html#biBEic00">[Eic00]</a> for a description of the algorithms underlying the functions in this section.</p>
<p><a id="X8036FA507A170DC4" name="X8036FA507A170DC4"></a></p>
<h5>7.4-1 TorsionSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TorsionSubgroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If the set of elements of finite order forms a subgroup, then we call it the <em>torsion subgroup</em>. This function determines the torsion subgroup of <var class="Arg">U</var>, if it exists, and returns fail otherwise. Note that a torsion subgroup does always exist if <var class="Arg">U</var> is nilpotent.</p>
<p><a id="X8082CD337972DC63" name="X8082CD337972DC63"></a></p>
<h5>7.4-2 NormalTorsionSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalTorsionSubgroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Each polycyclic groups has a unique largest finite normal subgroup. This function computes it for <var class="Arg">U</var>.</p>
<p><a id="X86D92DA17DCE22DD" name="X86D92DA17DCE22DD"></a></p>
<h5>7.4-3 IsTorsionFree</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTorsionFree</code>( <var class="Arg">U</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This function checks if <var class="Arg">U</var> is torsion free. It returns true or false.</p>
<p><a id="X819058217B4F3DC0" name="X819058217B4F3DC0"></a></p>
<h5>7.4-4 FiniteSubgroupClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FiniteSubgroupClasses</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>There exist only finitely many conjugacy classes of finite subgroups in a polycyclic group <var class="Arg">U</var> and this function can be used to compute them. The algorithm underlying this function proceeds by working down a normal series of <var class="Arg">U</var> with elementary or free abelian factors. The following function can be used to give the algorithm a specific series.</p>
<p><a id="X7E7C32EA81A297B6" name="X7E7C32EA81A297B6"></a></p>
<h5>7.4-5 FiniteSubgroupClassesBySeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FiniteSubgroupClassesBySeries</code>( <var class="Arg">U</var>, <var class="Arg">pcps</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ExamplesOfSomePcpGroups(15);</span>
Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">TorsionSubgroup(G);</span>
Pcp-group with orders [ 5, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">NormalTorsionSubgroup(G);</span>
Pcp-group with orders [ 5, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTorsionFree(G);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">FiniteSubgroupClasses(G);</span>
[ Pcp-group with orders [ 5, 2 ]^G,
Pcp-group with orders [ 2 ]^G,
Pcp-group with orders [ 5 ]^G,
Pcp-group with orders [ ]^G ]
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralPcpGroup( 0 );</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">TorsionSubgroup(G);</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">NormalTorsionSubgroup(G);</span>
Pcp-group with orders [ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTorsionFree(G);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">FiniteSubgroupClasses(G);</span>
[ Pcp-group with orders [ 2 ]^G,
Pcp-group with orders [ 2 ]^G,
Pcp-group with orders [ ]^G ]
</pre></div>
<p><a id="X7D9F737F80F6E396" name="X7D9F737F80F6E396"></a></p>
<h4>7.5 <span class="Heading">Subgroups of finite index and maximal subgroups</span></h4>
<p>Here we outline functions to determine various types of subgroups of finite index in polycyclic groups. Again, see <a href="chapBib.html#biBEic00">[Eic00]</a> for a description of the algorithms underlying the functions in this section. Also, we refer to <a href="chapBib.html#biBLo99">[Lo98a]</a> for an alternative approach.</p>
<p><a id="X87D62D497A8715FB" name="X87D62D497A8715FB"></a></p>
<h5>7.5-1 MaximalSubgroupClassesByIndex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalSubgroupClassesByIndex</code>( <var class="Arg">U</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Each maximal subgroup of a polycyclic group <var class="Arg">U</var> has <var class="Arg">p</var>-power index for some prime <var class="Arg">p</var>. This function can be used to determine the conjugacy classes of all maximal subgroups of <var class="Arg">p</var>-power index for a given prime <var class="Arg">p</var>.</p>
<p><a id="X7800133F81BC7674" name="X7800133F81BC7674"></a></p>
<h5>7.5-2 LowIndexSubgroupClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowIndexSubgroupClasses</code>( <var class="Arg">U</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>There are only finitely many subgroups of a given index in a polycyclic group <var class="Arg">U</var>. This function computes conjugacy classes of all subgroups of index <var class="Arg">n</var> in <var class="Arg">U</var>.</p>
<p><a id="X7F7067C77F2DC32C" name="X7F7067C77F2DC32C"></a></p>
<h5>7.5-3 LowIndexNormalSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowIndexNormalSubgroups</code>( <var class="Arg">U</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This function computes the normal subgroups of index <var class="Arg">n</var> in <var class="Arg">U</var>.</p>
<p><a id="X85A5BC447D83175F" name="X85A5BC447D83175F"></a></p>
<h5>7.5-4 NilpotentByAbelianNormalSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NilpotentByAbelianNormalSubgroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a normal subgroup <var class="Arg">N</var> of finite index in <var class="Arg">U</var> such that <var class="Arg">N</var> is nilpotent-by-abelian. Such a subgroup exists in every polycyclic group and this function computes such a subgroup using LowIndexNormal. However, we note that this function is not very efficient and the function NilpotentByAbelianByFiniteSeries may well be more efficient on this task.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ExamplesOfSomePcpGroups(2);</span>
Pcp-group with orders [ 0, 0, 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalSubgroupClassesByIndex( G, 61 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">max := List( last, Representative );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( max, x -> Index( G, x ) );</span>
[ 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
61, 61, 61, 61, 61, 61, 226981 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">LowIndexSubgroupClasses( G, 61 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">low := List( last, Representative );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( low, x -> Index( G, x ) );</span>
[ 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61,
61, 61, 61, 61, 61, 61 ]
</pre></div>
<p><a id="X785E0E877AB1D549" name="X785E0E877AB1D549"></a></p>
<h4>7.6 <span class="Heading">Further attributes for pcp-groups based on the Fitting subgroup</span></h4>
<p>In this section we provide a variety of other attributes for pcp-groups. Most of the methods below are based or related to the Fitting subgroup of the given group. We refer to <a href="chapBib.html#biBEic01">[Eic01b]</a> for a description of the underlying methods.</p>
<p><a id="X780552B57C30DD8F" name="X780552B57C30DD8F"></a></p>
<h5>7.6-1 FittingSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FittingSubgroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the Fitting subgroup of <var class="Arg">U</var>; that is, the largest nilpotent normal subgroup of <var class="Arg">U</var>.</p>
<p><a id="X86BD63DC844731DF" name="X86BD63DC844731DF"></a></p>
<h5>7.6-2 IsNilpotentByFinite</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNilpotentByFinite</code>( <var class="Arg">U</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>checks whether the Fitting subgroup of <var class="Arg">U</var> has finite index.</p>
<p><a id="X847ABE6F781C7FE8" name="X847ABE6F781C7FE8"></a></p>
<h5>7.6-3 Centre</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Centre</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the centre of <var class="Arg">U</var>.</p>
<p><a id="X861C36368435EB09" name="X861C36368435EB09"></a></p>
<h5>7.6-4 FCCentre</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FCCentre</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the FC-centre of <var class="Arg">U</var>; that is, the subgroup containing all elements having a finite conjugacy class in <var class="Arg">U</var>.</p>
<p><a id="X7E75E2BC806746AC" name="X7E75E2BC806746AC"></a></p>
<h5>7.6-5 PolyZNormalSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PolyZNormalSubgroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a normal subgroup <var class="Arg">N</var> of finite index in <var class="Arg">U</var>, such that <var class="Arg">N</var> has a polycyclic series with infinite factors only.</p>
<p><a id="X86800BF783E30D4A" name="X86800BF783E30D4A"></a></p>
<h5>7.6-6 NilpotentByAbelianByFiniteSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NilpotentByAbelianByFiniteSeries</code>( <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a normal series <span class="SimpleMath">1 ≤ F ≤ A ≤ U</span> such that <span class="SimpleMath">F</span> is nilpotent, <span class="SimpleMath">A/F</span> is abelian and <span class="SimpleMath">U/A</span> is finite. This series is computed using the Fitting subgroup and the centre of the Fitting factor.</p>
<p><a id="X878DBDC77CCA4F7E" name="X878DBDC77CCA4F7E"></a></p>
<h4>7.7 <span class="Heading">Functions for nilpotent groups</span></h4>
<p>There are (very few) functions which are available for nilpotent groups only. First, there are the different central series. These are available for all groups, but for nilpotent groups they terminate and provide series through the full group. Secondly, the determination of a minimal generating set is available for nilpotent groups only.</p>
<p><a id="X81D15723804771E2" name="X81D15723804771E2"></a></p>
<h5>7.7-1 MinimalGeneratingSet</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalGeneratingSet</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ExamplesOfSomePcpGroups(14);</span>
Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 0, 5, 5, 4, 0, 6,
5, 5, 4, 0, 10, 6 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNilpotent(G);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">PcpsBySeries( LowerCentralSeriesOfGroup(G));</span>
[ Pcp [ g1, g2 ] with orders [ 0, 0 ],
Pcp [ g3 ] with orders [ 0 ],
Pcp [ g4 ] with orders [ 0 ],
Pcp [ g5 ] with orders [ 0 ],
Pcp [ g6, g7 ] with orders [ 0, 0 ],
Pcp [ g8 ] with orders [ 0 ],
Pcp [ g9, g10 ] with orders [ 0, 0 ],
Pcp [ g11, g12, g13 ] with orders [ 5, 4, 0 ],
Pcp [ g14, g15, g16, g17, g18 ] with orders [ 5, 5, 4, 0, 6 ],
Pcp [ g19, g20, g21, g22, g23, g24 ] with orders [ 5, 5, 4, 0, 10, 6 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PcpsBySeries( UpperCentralSeriesOfGroup(G));</span>
[ Pcp [ g1, g2 ] with orders [ 0, 0 ],
Pcp [ g3 ] with orders [ 0 ],
Pcp [ g4 ] with orders [ 0 ],
Pcp [ g5 ] with orders [ 0 ],
Pcp [ g6, g7 ] with orders [ 0, 0 ],
Pcp [ g8 ] with orders [ 0 ],
Pcp [ g9, g10 ] with orders [ 0, 0 ],
Pcp [ g11, g12, g13 ] with orders [ 5, 4, 0 ],
Pcp [ g14, g15, g16, g17, g18 ] with orders [ 5, 5, 4, 0, 6 ],
Pcp [ g19, g20, g21, g22, g23, g24 ] with orders [ 5, 5, 4, 0, 10, 6 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalGeneratingSet(G);</span>
[ g1, g2 ]
</pre></div>
<p><a id="X8640F9D47A1F7434" name="X8640F9D47A1F7434"></a></p>
<h4>7.8 <span class="Heading">Random methods for pcp-groups</span></h4>
<p>Below we introduce a function which computes orbit and stabilizer using a random method. This function tries to approximate the orbit and the stabilizer, but the returned orbit or stabilizer may be incomplete. This function is used in the random methods to compute normalizers and centralizers. Note that deterministic methods for these purposes are also available.</p>
<p><a id="X80AEE73E7D639699" name="X80AEE73E7D639699"></a></p>
<h5>7.8-1 RandomCentralizerPcpGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomCentralizerPcpGroup</code>( <var class="Arg">U</var>, <var class="Arg">g</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">‣ RandomCentralizerPcpGroup</code>( <var class="Arg">U</var>, <var class="Arg">V</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">‣ RandomNormalizerPcpGroup</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralPcpGroup(0);</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">mats := [[[-1, 0],[0,1]], [[1,1],[0,1]]];</span>
[ [ [ -1, 0 ], [ 0, 1 ] ], [ [ 1, 1 ], [ 0, 1 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">pcp := Pcp(G);</span>
Pcp [ g1, g2 ] with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RandomPcpOrbitStabilizer( [1,0], pcp, mats, OnRight ).stab;</span>
#I Orbit longer than limit: exiting.
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g := Igs(G)[1];</span>
g1
<span class="GAPprompt">gap></span> <span class="GAPinput">RandomCentralizerPcpGroup( G, g );</span>
#I Stabilizer not increasing: exiting.
Pcp-group with orders [ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Igs(last);</span>
[ g1 ]
</pre></div>
<p><a id="X824142B784453DB9" name="X824142B784453DB9"></a></p>
<h4>7.9 <span class="Heading">Non-abelian tensor product and Schur extensions</span></h4>
<p><a id="X79EF28D9845878C9" name="X79EF28D9845878C9"></a></p>
<h5>7.9-1 SchurExtension</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SchurExtension</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a polycyclic group with a polycyclic generating sequence consisting of <span class="SimpleMath">n</span> elements. This function computes the largest central extension <var class="Arg">H</var> of <var class="Arg">G</var> such that <var class="Arg">H</var> is generated by <span class="SimpleMath">n</span> elements. If <span class="SimpleMath">F/R</span> is the underlying polycyclic presentation for <var class="Arg">G</var>, then <var class="Arg">H</var> is isomorphic to <span class="SimpleMath">F/[R,F]</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralPcpGroup( 0 );</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Centre( G );</span>
Pcp-group with orders [ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">H := SchurExtension( G );</span>
Pcp-group with orders [ 2, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Centre( H );</span>
Pcp-group with orders [ 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">H/Centre(H);</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Subgroup( H, [H.1,H.2] ) = H;</span>
true
</pre></div>
<p><a id="X84B60EC978A9A05E" name="X84B60EC978A9A05E"></a></p>
<h5>7.9-2 SchurExtensionEpimorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SchurExtensionEpimorphism</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the projection from the Schur extension <span class="SimpleMath">G^*</span> of <var class="Arg">G</var> onto <var class="Arg">G</var>. See the function <code class="code">SchurExtension</code>. The kernel of this epimorphism is the direct product of the Schur multiplicator of <var class="Arg">G</var> and a direct product of <span class="SimpleMath">n</span> copies of <span class="SimpleMath">ℤ</span> where <span class="SimpleMath">n</span> is the number of generators in the polycyclic presentation for <var class="Arg">G</var>. The Schur multiplicator is the intersection of the kernel and the derived group of the source. See also the function <code class="code">SchurCover</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gl23 := Range( IsomorphismPcpGroup( GL(2,3) ) );</span>
Pcp-group with orders [ 2, 3, 2, 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SchurExtensionEpimorphism( gl23 );</span>
[ g1, g2, g3, g4, g5, g6, g7, g8, g9, g10 ] -> [ g1, g2, g3, g4, g5,
id, id, id, id, id ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Kernel( last );</span>
Pcp-group with orders [ 0, 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariantsMultiplier( gl23 );</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( Kernel(epi), DerivedSubgroup( Source(epi) ) );</span>
[ ]
</pre></div>
<p>There is a crossed pairing from <var class="Arg">G</var> into <span class="SimpleMath">(G^*)'</span> which can be defined via this epimorphism:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralPcpGroup(0);</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">epi := SchurExtensionEpimorphism( G );</span>
[ g1, g2, g3, g4 ] -> [ g1, g2, id, id ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesRepresentative( epi, G.1 );</span>
g1
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesRepresentative( epi, G.2 );</span>
g2
<span class="GAPprompt">gap></span> <span class="GAPinput">Comm( last, last2 );</span>
g2^-2*g4
</pre></div>
<p><a id="X7DD1E37987612042" name="X7DD1E37987612042"></a></p>
<h5>7.9-3 SchurCover</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SchurCover</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes a Schur covering group of the polycyclic group <var class="Arg">G</var>. A Schur covering is a largest central extension <var class="Arg">H</var> of <var class="Arg">G</var> such that the kernel <var class="Arg">M</var> of the projection of <var class="Arg">H</var> onto <var class="Arg">G</var> is contained in the commutator subgroup of <var class="Arg">H</var>.</p>
<p>If <var class="Arg">G</var> is given by a presentation <span class="SimpleMath">F/R</span>, then <var class="Arg">M</var> is isomorphic to the subgroup <span class="SimpleMath">R ∩ [F,F] / [R,F]</span>. Let <span class="SimpleMath">C</span> be a complement to <span class="SimpleMath">R ∩ [F,F] / [R,F]</span> in <span class="SimpleMath">R/[R,F]</span>. Then <span class="SimpleMath">F/C</span> is isomorphic to <var class="Arg">H</var> and <span class="SimpleMath">R/C</span> is isomorphic to <var class="Arg">M</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := AbelianPcpGroup( 3 );</span>
Pcp-group with orders [ 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ext := SchurCover( G );</span>
Pcp-group with orders [ 0, 0, 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Centre( ext );</span>
Pcp-group with orders [ 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup( DerivedSubgroup( ext ), last );</span>
true
</pre></div>
<p><a id="X792BC39D7CEB1D27" name="X792BC39D7CEB1D27"></a></p>
<h5>7.9-4 AbelianInvariantsMultiplier</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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 G.</p>
<p>Note that the Schur multiplicator of a polycyclic group is a finitely generated abelian group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralPcpGroup( 0 );</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DirectProduct( G, AbelianPcpGroup( 2 ) );</span>
Pcp-group with orders [ 0, 0, 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariantsMultiplier( last );</span>
[ 0, 2, 2, 2, 2 ]
</pre></div>
<p><a id="X822ED5978647C93B" name="X822ED5978647C93B"></a></p>
<h5>7.9-5 NonAbelianExteriorSquareEpimorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonAbelianExteriorSquareEpimorphism</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the epimorphism of the non-abelian exterior square of a polycyclic group <var class="Arg">G</var> onto the derived group of <var class="Arg">G</var>. The non-abelian exterior square can be defined as the derived subgroup of a Schur cover of <var class="Arg">G</var>. The isomorphism type of the non-abelian exterior square is unique despite the fact that the isomorphism type of a Schur cover of a polycyclic groups need not be unique. The derived group of a Schur cover has a natural projection onto the derived group of <var class="Arg">G</var> which is what the function returns.</p>
<p>The kernel of the epimorphism is isomorphic to the Schur multiplicator of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ExamplesOfSomePcpGroups( 3 );</span>
Pcp-group with orders [ 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DirectProduct( G,G );</span>
Pcp-group with orders [ 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariantsMultiplier( G );</span>
[ [ 0, 1 ], [ 2, 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">epi := NonAbelianExteriorSquareEpimorphism( G );</span>
[ g2^-2*g5, g4^-2*g10, g6, g7, g8, g9 ] -> [ g2^-2, g4^-2, id, id, id, id ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Kernel( epi );</span>
Pcp-group with orders [ 0, 2, 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Collected( AbelianInvariants( last ) );</span>
[ [ 0, 1 ], [ 2, 3 ] ]
</pre></div>
<p><a id="X8739CD4686301A0E" name="X8739CD4686301A0E"></a></p>
<h5>7.9-6 NonAbelianExteriorSquare</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonAbelianExteriorSquare</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the non-abelian exterior square of a polycyclic group <var class="Arg">G</var>. See the explanation for <code class="code">NonAbelianExteriorSquareEpimorphism</code>. The natural projection of the non-abelian exterior square onto the derived group of <var class="Arg">G</var> is stored in the component <code class="code">!.epimorphism</code>.</p>
<p>There is a crossed pairing from <span class="SimpleMath">G× G</span> into <span class="SimpleMath">G∧ G</span>. See the function <code class="code">SchurExtensionEpimorphism</code> for details. The crossed pairing is stored in the component <code class="code">!.crossedPairing</code>. This is the crossed pairing <span class="SimpleMath">λ</span> in <a href="chapBib.html#biBEickNickel07">[EN08]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralPcpGroup(0);</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GwG := NonAbelianExteriorSquare( G );</span>
Pcp-group with orders [ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">lambda := GwG!.crossedPairing;</span>
function( g, h ) ... end
<span class="GAPprompt">gap></span> <span class="GAPinput">lambda( G.1, G.2 );</span>
g2^2*g4^-1
</pre></div>
<p><a id="X86553D7B7DABF38F" name="X86553D7B7DABF38F"></a></p>
<h5>7.9-7 NonAbelianTensorSquareEpimorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonAbelianTensorSquareEpimorphism</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns for a polycyclic group <var class="Arg">G</var> the projection of the non-abelian tensor square <span class="SimpleMath">G⊗ G</span> onto the non-abelian exterior square <span class="SimpleMath">G∧ G</span>. The range of that epimorphism has the component <code class="code">!.epimorphism</code> set to the projection of the non-abelian exterior square onto the derived group of <var class="Arg">G</var>. See also the function <code class="code">NonAbelianExteriorSquare</code>.</p>
<p>With the result of this function one can compute the groups in the commutative diagram at the beginning of the paper <a href="chapBib.html#biBEickNickel07">[EN08]</a>. The kernel of the returned epimorphism is the group <span class="SimpleMath">∇(G)</span>. The kernel of the composition of this epimorphism and the above mention projection onto <span class="SimpleMath">G'</span> is the group <span class="SimpleMath">J(G)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralPcpGroup(0);</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DirectProduct(G,G);</span>
Pcp-group with orders [ 2, 0, 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha := NonAbelianTensorSquareEpimorphism( G );</span>
[ g9*g25^-1, g10*g26^-1, g11*g27, g12*g28, g13*g29, g14*g30, g15, g16,
g17,
g18, g19, g20, g21, g22, g23, g24 ] -> [ g2^-2*g6, g4^-2*g12, g8,
g9, g10,
g11, id, id, id, id, id, id, id, id, id, id ]
<span class="GAPprompt">gap></span> <span class="GAPinput">gamma := Range( alpha )!.epimorphism;</span>
[ g2^-2*g6, g4^-2*g12, g8, g9, g10, g11 ] -> [ g2^-2, g4^-2, id, id,
id, id ]
<span class="GAPprompt">gap></span> <span class="GAPinput">JG := Kernel( alpha * gamma );</span>
Pcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Image( alpha, JG );</span>
Pcp-group with orders [ 2, 2, 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariantsMultiplier( G );</span>
[ [ 2, 4 ] ]
</pre></div>
<p><a id="X7C0DF7C97F78C666" name="X7C0DF7C97F78C666"></a></p>
<h5>7.9-8 NonAbelianTensorSquare</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonAbelianTensorSquare</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes for a polycyclic group <var class="Arg">G</var> the non-abelian tensor square <span class="SimpleMath">G⊗ G</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := AlternatingGroup( IsPcGroup, 4 );</span>
<pc group of size 12 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">PcGroupToPcpGroup( G );</span>
Pcp-group with orders [ 3, 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">NonAbelianTensorSquare( last );</span>
Pcp-group with orders [ 2, 2, 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PcpGroupToPcGroup( last );</span>
<pc group of size 24 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">DirectFactorsOfGroup( last );</span>
[ Group([ f1, f2, f3 ]), Group([ f4 ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( last, Size );</span>
[ 8, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup( last2[1] );</span>
[ 8, 4 ] # the quaternion group of Order 8
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralPcpGroup( 0 );</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ten := NonAbelianTensorSquare( G );</span>
Pcp-group with orders [ 0, 2, 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAbelian( ten );</span>
true
</pre></div>
<p><a id="X7AE75EC1860FFE7A" name="X7AE75EC1860FFE7A"></a></p>
<h5>7.9-9 NonAbelianExteriorSquarePlusEmbedding</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonAbelianExteriorSquarePlusEmbedding</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an embedding from the non-abelian exterior square <span class="SimpleMath">G∧ G</span> into an extensions of <span class="SimpleMath">G∧ G</span> by <span class="SimpleMath">G× G</span>. For the significance of the group see the paper <a href="chapBib.html#biBEickNickel07">[EN08]</a>. The range of the epimorphism is the group <span class="SimpleMath">τ(G)</span> in that paper.</p>
<p><a id="X7D96C84E87925B0F" name="X7D96C84E87925B0F"></a></p>
<h5>7.9-10 NonAbelianTensorSquarePlusEpimorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonAbelianTensorSquarePlusEpimorphism</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an epimorphisms of <span class="SimpleMath">ν(G)</span> onto <span class="SimpleMath">τ(G)</span>. The group <span class="SimpleMath">ν(G)</span> is an extension of the non-abelian tensor square <span class="SimpleMath">G⊗ G</span> of <span class="SimpleMath">G</span> by <span class="SimpleMath">G× G</span>. The group <span class="SimpleMath">τ(G)</span> is an extension of the non-abelian exterior square <span class="SimpleMath">G∧ G</span> by <span class="SimpleMath">G× G</span>. For details see <a href="chapBib.html#biBEickNickel07">[EN08]</a>.</p>
<p><a id="X8746533787C4E8BC" name="X8746533787C4E8BC"></a></p>
<h5>7.9-11 NonAbelianTensorSquarePlus</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonAbelianTensorSquarePlus</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the group <span class="SimpleMath">ν(G)</span> in <a href="chapBib.html#biBEickNickel07">[EN08]</a>.</p>
<p><a id="X78F9184078B2761A" name="X78F9184078B2761A"></a></p>
<h5>7.9-12 WhiteheadQuadraticFunctor</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadQuadraticFunctor</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns Whitehead's universal quadratic functor of <span class="SimpleMath">G</span>, see <a href="chapBib.html#biBEickNickel07">[EN08]</a> for a description.</p>
<p><a id="X7D3023697BA5CE5A" name="X7D3023697BA5CE5A"></a></p>
<h4>7.10 <span class="Heading">Schur covers</span></h4>
<p>This section contains a function to determine the Schur covers of a finite <span class="SimpleMath">p</span>-group up to isomorphism.</p>
<p><a id="X7D90B44E7B96AFF1" name="X7D90B44E7B96AFF1"></a></p>
<h5>7.10-1 SchurCovers</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SchurCovers</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a finite <span class="SimpleMath">p</span>-group defined as a pcp group. This function returns a complete and irredundant set of isomorphism types of Schur covers of <var class="Arg">G</var>. The algorithm implements a method of Nickel's Phd Thesis.</p>
<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap6.html">[Previous Chapter]</a> <a href="chap8.html">[Next Chapter]</a> </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="chapA.html">A</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
|
