39:
28:
47:
433:
259:
504:
653:. The modular maximal-cyclic group of order 2 always has nilpotency class 2. This makes the modular maximal-cyclic group less interesting, since most groups of order
621:
582:
543:
304:
128:, this group is called the "semidihedral group". Dummit and Foote refer to it as the "quasidihedral group"; we adopt that name in this article. All give the same
669:
312:
138:
38:
822:
780:
265:
The other non-abelian 2-group with cyclic subgroup of index 2 is not given a special name in either text, but referred to as just
27:
101:
626:
The generalized quaternion group, the dihedral group, and the quasidihedral group of order 2 all have nilpotency class
46:
461:
108:. One of the remaining two groups is often considered particularly important, since it is an example of a 2-group of
275:(2). When this group has order 16, Dummit and Foote refer to this group as the "modular group of order 16", as its
451:> of order 2. Such a non-abelian semidirect product is uniquely determined by an element of order 2 in the
849:
71:
97:
86:
664:
The generalized quaternion, the dihedral, and the quasidihedral group are the only 2-groups whose
456:
440:
129:
623:, corresponding to the dihedral group, the quasidihedral, and the modular maximal-cyclic group.
587:
548:
509:
818:
810:
776:
121:
82:
67:
665:
832:
802:
282:
828:
798:
790:
113:
452:
279:. In this article this group will be called the modular maximal-cyclic group of order
105:
843:
733:
630:â 1, and are the only isomorphism classes of groups of order 2 with nilpotency class
276:
109:
677:
673:
428:{\displaystyle \langle r,s\mid r^{2^{n-1}}=s^{2}=1,\ srs=r^{2^{n-2}+1}\rangle \,\!}
254:{\displaystyle \langle r,s\mid r^{2^{n-1}}=s^{2}=1,\ srs=r^{2^{n-2}-1}\rangle \,\!}
90:
31:
55:
17:
661:
have nilpotency class 2 and have proven difficult to understand directly.
93:
650:
643:
75:
45:
37:
26:
688:
The Sylow 2-subgroups of the following groups are quasidihedral:
42:
Cayley graph of the modular maximal-cyclic group of order 16
590:
551:
512:
464:
315:
285:
141:
642:â 1 were the beginning of the classification of all
81:greater than or equal to 4, there are exactly four
615:
576:
537:
498:
427:
298:
253:
120:, this group is called a "Quasidiedergruppe". In
439:Both these two groups and the dihedral group are
424:
250:
499:{\displaystyle \mathbb {Z} /2^{n-1}\mathbb {Z} }
50:Cayley graph of the dihedral group of order 16
506:and there are precisely three such elements,
8:
420:
316:
246:
142:
595:
589:
556:
550:
517:
511:
492:
491:
479:
470:
466:
465:
463:
423:
400:
395:
361:
340:
335:
314:
290:
284:
249:
226:
221:
187:
166:
161:
140:
680:, with quasidihedral Sylow 2-subgroups.
447:> of order 2 with a cyclic group <
34:of the quasidihedral group of order 16
775:(3 ed.). Wiley. pp. 71â72.
7:
74:a power of 2. For every positive
25:
771:Dummit, D. S.; Foote, R. (2004).
670:AlperinâBrauerâGorenstein theorem
277:lattice of subgroups is modular
1:
817:. Chelsea. pp. 188â195.
797:. Springer. pp. 90â93.
102:generalized quaternion group
100:2. Two are well known, the
866:
634:â 1. The groups of order
616:{\displaystyle 2^{n-2}+1}
577:{\displaystyle 2^{n-2}-1}
538:{\displaystyle 2^{n-1}-1}
110:maximal nilpotency class
89:of order 2 which have a
306:. Its presentation is:
676:, and to a degree the
617:
578:
539:
500:
443:of a cyclic group <
429:
300:
255:
51:
43:
35:
638:and nilpotency class
618:
579:
540:
501:
430:
301:
299:{\displaystyle 2^{n}}
256:
60:quasi-dihedral groups
49:
41:
30:
588:
549:
510:
462:
313:
283:
139:
64:semi-dihedral groups
441:semidirect products
83:isomorphism classes
668:has index 4. The
613:
574:
535:
496:
425:
296:
251:
68:non-abelian groups
52:
44:
36:
378:
204:
122:Daniel Gorenstein
16:(Redirected from
857:
836:
806:
795:Endliche Gruppen
786:
773:Abstract Algebra
666:derived subgroup
622:
620:
619:
614:
606:
605:
583:
581:
580:
575:
567:
566:
544:
542:
541:
536:
528:
527:
505:
503:
502:
497:
495:
490:
489:
474:
469:
434:
432:
431:
426:
419:
418:
411:
410:
376:
366:
365:
353:
352:
351:
350:
305:
303:
302:
297:
295:
294:
260:
258:
257:
252:
245:
244:
237:
236:
202:
192:
191:
179:
178:
177:
176:
132:for this group:
118:Endliche Gruppen
21:
865:
864:
860:
859:
858:
856:
855:
854:
840:
839:
825:
809:
789:
783:
770:
767:
755:
746:
739:
724:
715:
704:
695:
686:
672:classifies the
591:
586:
585:
552:
547:
546:
513:
508:
507:
475:
460:
459:
396:
391:
357:
336:
331:
311:
310:
286:
281:
280:
274:
222:
217:
183:
162:
157:
137:
136:
114:Bertram Huppert
85:of non-abelian
62:, also called
23:
22:
15:
12:
11:
5:
863:
861:
853:
852:
842:
841:
838:
837:
823:
811:Gorenstein, D.
807:
787:
781:
766:
763:
762:
761:
751:
744:
741:
737:
730:
720:
713:
710:
700:
693:
685:
682:
612:
609:
604:
601:
598:
594:
573:
570:
565:
562:
559:
555:
534:
531:
526:
523:
520:
516:
494:
488:
485:
482:
478:
473:
468:
453:group of units
437:
436:
422:
417:
414:
409:
406:
403:
399:
394:
390:
387:
384:
381:
375:
372:
369:
364:
360:
356:
349:
346:
343:
339:
334:
330:
327:
324:
321:
318:
293:
289:
270:
263:
262:
248:
243:
240:
235:
232:
229:
225:
220:
216:
213:
210:
207:
201:
198:
195:
190:
186:
182:
175:
172:
169:
165:
160:
156:
153:
150:
147:
144:
106:dihedral group
66:, are certain
24:
14:
13:
10:
9:
6:
4:
3:
2:
862:
851:
850:Finite groups
848:
847:
845:
834:
830:
826:
824:0-8284-0301-5
820:
816:
815:Finite Groups
812:
808:
804:
800:
796:
792:
788:
784:
782:9780471433347
778:
774:
769:
768:
764:
759:
754:
750:
742:
735:
734:Mathieu group
731:
728:
723:
719:
711:
708:
703:
699:
691:
690:
689:
683:
681:
679:
678:finite groups
675:
674:simple groups
671:
667:
662:
660:
656:
652:
648:
646:
641:
637:
633:
629:
624:
610:
607:
602:
599:
596:
592:
571:
568:
563:
560:
557:
553:
532:
529:
524:
521:
518:
514:
486:
483:
480:
476:
471:
458:
454:
450:
446:
442:
415:
412:
407:
404:
401:
397:
392:
388:
385:
382:
379:
373:
370:
367:
362:
358:
354:
347:
344:
341:
337:
332:
328:
325:
322:
319:
309:
308:
307:
291:
287:
278:
273:
268:
241:
238:
233:
230:
227:
223:
218:
214:
211:
208:
205:
199:
196:
193:
188:
184:
180:
173:
170:
167:
163:
158:
154:
151:
148:
145:
135:
134:
133:
131:
127:
126:Finite Groups
123:
119:
115:
111:
107:
103:
99:
95:
92:
88:
84:
80:
77:
73:
69:
65:
61:
57:
48:
40:
33:
29:
19:
18:Quasidihedral
814:
794:
772:
757:
752:
748:
726:
721:
717:
706:
701:
697:
687:
663:
658:
654:
644:
639:
635:
631:
627:
625:
448:
444:
438:
271:
266:
264:
130:presentation
125:
117:
78:
63:
59:
53:
32:Cayley graph
791:Huppert, B.
56:mathematics
765:References
760:⥠3 mod 4.
729:⥠1 mod 4,
709:⥠3 mod 4,
657:for large
600:−
569:−
561:−
530:−
522:−
484:−
421:⟩
405:−
345:−
329:∣
317:⟨
247:⟩
239:−
231:−
171:−
155:∣
143:⟨
124:'s text,
844:Category
813:(1980).
793:(1967).
684:Examples
116:'s text
104:and the
94:subgroup
833:0569209
803:0224703
651:coclass
647:-groups
455:of the
76:integer
831:
821:
801:
779:
756:) for
725:) for
705:) for
584:, and
377:
203:
112:. In
91:cyclic
87:groups
58:, the
98:index
72:order
819:ISBN
777:ISBN
732:the
649:via
457:ring
269:or M
712:PSU
692:PSL
96:of
70:of
54:In
846::
829:MR
827:.
799:MR
743:GL
738:11
545:,
835:.
805:.
785:.
758:q
753:q
749:F
747:(
745:2
740:,
736:M
727:q
722:q
718:F
716:(
714:3
707:q
702:q
698:F
696:(
694:3
659:n
655:p
645:p
640:n
636:p
632:n
628:n
611:1
608:+
603:2
597:n
593:2
572:1
564:2
558:n
554:2
533:1
525:1
519:n
515:2
493:Z
487:1
481:n
477:2
472:/
467:Z
449:s
445:r
435:.
416:1
413:+
408:2
402:n
398:2
393:r
389:=
386:s
383:r
380:s
374:,
371:1
368:=
363:2
359:s
355:=
348:1
342:n
338:2
333:r
326:s
323:,
320:r
292:n
288:2
272:m
267:G
261:.
242:1
234:2
228:n
224:2
219:r
215:=
212:s
209:r
206:s
200:,
197:1
194:=
189:2
185:s
181:=
174:1
168:n
164:2
159:r
152:s
149:,
146:r
79:n
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
