144:
658:
636:
693:
293:
59:
268:
190:(via sending an element to conjugation by that element), although the converse need not hold: for example, the
48:
239:
52:
194:
of 8 elements is isomorphic to its automorphism group, but it is not complete. For a discussion, see (
213:
155:
21:
674:
398:
187:
44:
654:
632:
183:
86:
94:
650:
628:
297:
110:
191:
687:
159:
283:
151:
34:
363:
17:
98:
90:
448:
678:
477:
is a direct product. The proof follows directly from the definition:
441:
This can be restated in terms of elements and internal conditions: If
97:
is the identity automorphism, meaning the group is centerless, while
344:
is surjective and has an obvious section given by the inclusion of
529:
and this conjugation must be equal to conjugation by some element
286:
and exact sequences can be given in a natural way: The action of
136:, the group has a non-trivial center, while for the case
664:(chapter 7, in particular theorems 7.15 and 7.17).
47:, and it is centerless; that is, it has a trivial
261:, is a complete group then the extension splits:
679:How tall is the automorphism tower of a group?
8:
438:is trivial, and so the product is direct.
58:Equivalently, a group is complete if the
195:
647:An introduction to the theory of groups
101:implies it has no outer automorphisms.
306:, gives rise to a group homomorphism,
93:implies that only conjugation by the
7:
338:has trivial center the homomorphism
623:Robinson, Derek John Scott (1996),
512:then it induces an automorphism of
14:
625:A course in the theory of groups
212:is a group extension given as a
1:
202:Extensions of complete groups
168:, the automorphism group of
150:The automorphism group of a
182:A complete group is always
122:, are complete except when
710:
645:Rotman, Joseph J. (1994),
49:outer automorphism group
541:. Then conjugation by
109:As an example, all the
447:is a normal, complete
483:is centerless giving
267:is isomorphic to the
694:Properties of groups
649:, Berlin, New York:
627:, Berlin, New York:
518:by conjugation, but
418:, but the action of
214:short exact sequence
206:Assume that a group
73:(sending an element
573:and every element,
547:is the identity on
156:almost simple group
675:Joel David Hamkins
399:semidirect product
188:automorphism group
145:outer automorphism
79:to conjugation by
660:978-0-387-94285-8
638:978-0-387-94461-6
506:is an element of
356:. The kernel of
255:. If the kernel,
198:, section 13.5).
701:
663:
641:
612:
595:
584:
578:
572:
558:
552:
546:
540:
534:
528:
517:
511:
505:
500:is trivial. If
499:
482:
476:
456:
446:
437:
423:
417:
396:
390:
384:
378:
361:
355:
349:
343:
337:
331:
323:
305:
291:
282:. A proof using
281:
266:
260:
254:
248:, and quotient,
247:
234:
211:
173:
167:
142:
135:
129:}. For the case
128:
121:
111:symmetric groups
95:identity element
84:
78:
72:
42:
28:
709:
708:
704:
703:
702:
700:
699:
698:
684:
683:
671:
661:
651:Springer-Verlag
644:
639:
629:Springer-Verlag
622:
619:
603:
597:
586:
580:
574:
566:
560:
554:
548:
542:
536:
530:
519:
513:
507:
501:
490:
484:
478:
467:
458:
452:
442:
431:
425:
419:
408:
402:
392:
386:
380:
372:
366:
357:
351:
345:
339:
333:
325:
307:
301:
298:normal subgroup
287:
272:
262:
256:
249:
243:
220:
207:
204:
180:
169:
163:
137:
130:
123:
120:
114:
107:
80:
74:
63:
38:
24:
12:
11:
5:
707:
705:
697:
696:
686:
685:
682:
681:
670:
669:External links
667:
666:
665:
659:
642:
637:
618:
615:
599:
562:
486:
463:
427:
404:
397:is at least a
368:
269:direct product
236:
235:
203:
200:
192:dihedral group
179:
176:
143:, there is an
116:
106:
103:
29:is said to be
13:
10:
9:
6:
4:
3:
2:
706:
695:
692:
691:
689:
680:
676:
673:
672:
668:
662:
656:
652:
648:
643:
640:
634:
630:
626:
621:
620:
616:
614:
611:
607:
602:
594:
590:
585:is a product
583:
577:
570:
565:
557:
551:
545:
539:
533:
526:
522:
516:
510:
504:
498:
494:
489:
481:
475:
471:
466:
461:
455:
450:
445:
439:
435:
430:
422:
416:
412:
407:
400:
395:
389:
383:
376:
371:
365:
360:
354:
348:
342:
336:
329:
322:
318:
314:
310:
304:
299:
295:
290:
285:
284:homomorphisms
279:
275:
270:
265:
259:
252:
246:
241:
232:
228:
224:
219:
218:
217:
215:
210:
201:
199:
197:
196:Robinson 1996
193:
189:
185:
177:
175:
174:is complete.
172:
166:
162:simple group
161:
157:
153:
148:
146:
140:
133:
126:
119:
112:
104:
102:
100:
96:
92:
88:
83:
77:
70:
66:
61:
56:
54:
50:
46:
41:
36:
32:
27:
23:
19:
646:
624:
609:
605:
600:
592:
588:
581:
575:
568:
563:
555:
549:
543:
537:
531:
524:
520:
514:
508:
502:
496:
492:
487:
479:
473:
469:
464:
459:
453:
443:
440:
433:
428:
420:
414:
410:
405:
393:
387:
381:
374:
369:
358:
352:
346:
340:
334:
327:
320:
316:
312:
308:
302:
288:
277:
273:
263:
257:
250:
244:
237:
230:
226:
222:
208:
205:
181:
170:
164:
158:; for a non-
152:simple group
149:
138:
131:
124:
117:
108:
99:surjectivity
81:
75:
68:
64:
57:
51:and trivial
39:
35:automorphism
30:
25:
15:
451:of a group
364:centralizer
294:conjugation
91:injectivity
87:isomorphism
60:conjugation
18:mathematics
617:References
495:) ∩
216:of groups
184:isomorphic
178:Properties
391:, and so
324:. Since
296:) on the
85:), is an
33:if every
688:Category
449:subgroup
311: :
105:Examples
31:complete
553:and so
457:, then
362:is the
186:to its
160:abelian
127:∈ {2, 6
657:
635:
559:is in
523:= Aut(
315:→ Aut(
240:kernel
154:is an
67:→ Aut(
53:center
579:, of
330:) = 1
238:with
233:′ ⟶ 1
62:map,
45:inner
22:group
655:ISBN
633:ISBN
472:) ×
413:) ⋊
332:and
326:Out(
319:) ≅
292:(by
221:1 ⟶
20:, a
596:in
535:of
462:= C
424:on
385:in
379:of
350:in
141:= 6
134:= 2
43:is
37:of
16:In
690::
677::
653:,
631:,
613:.
589:gn
556:gn
544:gn
401:,
300:,
276:×
271:,
242:,
229:⟶
225:⟶
147:.
113:,
89::
55:.
610:N
608:)
606:N
604:(
601:G
598:C
593:n
591:)
587:(
582:G
576:g
571:)
569:N
567:(
564:G
561:C
550:N
538:N
532:n
527:)
525:N
521:N
515:N
509:G
503:g
497:N
493:N
491:(
488:G
485:C
480:N
474:N
470:N
468:(
465:G
460:G
454:G
444:N
436:)
434:N
432:(
429:G
426:C
421:N
415:N
411:N
409:(
406:G
403:C
394:G
388:G
382:N
377:)
375:N
373:(
370:G
367:C
359:φ
353:G
347:N
341:φ
335:N
328:N
321:N
317:N
313:G
309:φ
303:N
289:G
280:′
278:G
274:N
264:G
258:N
253:′
251:G
245:N
231:G
227:G
223:N
209:G
171:G
165:G
139:n
132:n
125:n
118:n
115:S
82:g
76:g
71:)
69:G
65:G
40:G
26:G
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.