519:
25:
380:
211:
407:
281:
252:
560:
364:
318:) must be non-abelian simple, as inner automorphism groups are never non-trivial cyclic). All non-abelian simple groups are quasisimple.
42:
453:
108:
89:
553:
141:
518:
61:
46:
579:
68:
349:
584:
546:
445:
75:
368:
338:
163:
57:
35:
284:
155:
417:
360:
299:
134:
356:
355:
The quasisimple groups are often studied alongside the simple groups and groups related to their
322:
303:
82:
449:
386:
496:
459:
427:
422:
463:
330:
326:
257:
219:
530:
348:, and along with the minimal normal soluble subgroups generates a subgroup called the
573:
477:
334:
138:
526:
148:
122:
24:
292:
344:
The subgroup generated by the subnormal quasisimple subgroups is called the
311:
325:
quasisimple subgroups of a group control the structure of a finite
18:
298:
Equivalently, a group is quasisimple if it is equal to its
534:
389:
367:
of the quasisimple groups is nearly identical to the
260:
222:
166:
49:. Unsourced material may be challenged and removed.
478:http://mathworld.wolfram.com/QuasisimpleGroup.html
401:
275:
246:
205:
554:
314:by its center) is simple (and it follows Inn(
8:
561:
547:
388:
381:covering groups of the alternating groups
259:
221:
165:
109:Learn how and when to remove this message
206:{\displaystyle 1\to Z(E)\to E\to S\to 1}
489:
7:
515:
513:
383:are quasisimple but not simple, for
329:in much the same way as the minimal
47:adding citations to reliable sources
14:
517:
369:projective representation theory
23:
34:needs additional citations for
270:
264:
241:
229:
197:
191:
185:
182:
176:
170:
154:. In other words, there is a
1:
337:do, and so are given a name,
533:. You can help Knowledge by
440:Aschbacher, Michael (2000).
350:generalized Fitting subgroup
601:
512:
446:Cambridge University Press
402:{\displaystyle n\geq 5.}
304:inner automorphism group
529:-related article is a
403:
371:of the simple groups.
277:
248:
207:
404:
365:representation theory
278:
249:
208:
580:Properties of groups
387:
361:almost simple groups
276:{\displaystyle Z(E)}
258:
220:
164:
156:short exact sequence
43:improve this article
501:Finite group theory
442:Finite Group Theory
418:Almost simple group
357:automorphism groups
300:commutator subgroup
58:"Quasisimple group"
585:Group theory stubs
399:
273:
247:{\displaystyle E=}
244:
203:
542:
541:
291:and denotes the
142:central extension
129:(also known as a
127:quasisimple group
119:
118:
111:
93:
592:
563:
556:
549:
521:
514:
504:
497:I. Martin Isaacs
494:
467:
428:Semisimple group
423:Schur multiplier
408:
406:
405:
400:
331:normal subgroups
282:
280:
279:
274:
253:
251:
250:
245:
212:
210:
209:
204:
114:
107:
103:
100:
94:
92:
51:
27:
19:
600:
599:
595:
594:
593:
591:
590:
589:
570:
569:
568:
567:
510:
508:
507:
503:(2008), p. 272.
495:
491:
486:
474:
456:
439:
436:
414:
385:
384:
377:
327:insoluble group
256:
255:
218:
217:
162:
161:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
598:
596:
588:
587:
582:
572:
571:
566:
565:
558:
551:
543:
540:
539:
522:
506:
505:
488:
487:
485:
482:
481:
480:
473:
472:External links
470:
469:
468:
454:
435:
432:
431:
430:
425:
420:
413:
410:
398:
395:
392:
376:
373:
272:
269:
266:
263:
243:
240:
237:
234:
231:
228:
225:
214:
213:
202:
199:
196:
193:
190:
187:
184:
181:
178:
175:
172:
169:
131:covering group
117:
116:
31:
29:
22:
16:Covering group
15:
13:
10:
9:
6:
4:
3:
2:
597:
586:
583:
581:
578:
577:
575:
564:
559:
557:
552:
550:
545:
544:
538:
536:
532:
528:
523:
520:
516:
511:
502:
498:
493:
490:
483:
479:
476:
475:
471:
465:
461:
457:
455:0-521-78675-4
451:
447:
443:
438:
437:
433:
429:
426:
424:
421:
419:
416:
415:
411:
409:
396:
393:
390:
382:
374:
372:
370:
366:
362:
358:
353:
351:
347:
342:
340:
336:
335:soluble group
332:
328:
324:
319:
317:
313:
309:
305:
301:
296:
294:
290:
286:
267:
261:
238:
235:
232:
226:
223:
200:
194:
188:
179:
173:
167:
160:
159:
158:
157:
153:
150:
146:
143:
140:
136:
132:
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
535:expanding it
527:group theory
524:
509:
500:
492:
441:
378:
354:
345:
343:
333:of a finite
320:
315:
307:
297:
288:
283:denotes the
215:
151:
149:simple group
144:
130:
126:
120:
105:
99:October 2009
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
123:mathematics
574:Categories
464:0997.20001
434:References
293:commutator
216:such that
137:that is a
69:newspapers
394:≥
339:component
323:subnormal
198:→
192:→
186:→
171:→
412:See also
375:Examples
312:quotient
302:and its
254:, where
363:. The
310:) (its
139:perfect
133:) is a
83:scholar
462:
452:
359:, the
285:center
85:
78:
71:
64:
56:
525:This
484:Notes
346:layer
147:of a
135:group
90:JSTOR
76:books
531:stub
450:ISBN
379:The
321:The
306:Inn(
125:, a
62:news
460:Zbl
287:of
121:In
45:by
576::
499:,
458:.
448:.
444:.
397:5.
352:.
341:.
295:.
562:e
555:t
548:v
537:.
466:.
391:n
316:G
308:G
289:E
271:)
268:E
265:(
262:Z
242:]
239:E
236:,
233:E
230:[
227:=
224:E
201:1
195:S
189:E
183:)
180:E
177:(
174:Z
168:1
152:S
145:E
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
