191:
445:
states that the chief factors of a group are unique up to isomorphism, independent of the particular chief series they are constructed from In particular, the number of chief factors is an
109:
52:
Chief series can be thought of as breaking the group down into less complicated pieces, which may be used to characterize various qualities of the group.
386:
Finite groups always have a chief series, though infinite groups need not have a chief series. For example, the group of integers
60:
A chief series is a maximal normal series for a group. Equivalently, a chief series is a composition series of the group
587:
362:
442:
278:. In other words, a chief series may be thought of as "full" in the sense that no normal subgroup of
606:
465:
In abelian groups, chief series and composition series are identical, as all subgroups are normal.
402:, and so all of its subgroups are normal and cyclic as well. Supposing there exists a chief series
370:
446:
366:
227:
441:
When a chief series for a group exists, it is generally not unique. However, a form of the
8:
76:
29:
186:{\displaystyle 1=N_{0}\subseteq N_{1}\subseteq N_{2}\subseteq \cdots \subseteq N_{n}=G,}
309:
65:
36:
583:
454:
558:
45:
17:
549:
Lafuente, J. (November 1978). "Homomorphs and formations of given derived class".
88:
390:
with addition as the operation does not have a chief series. To see this, note
197:
39:, though the two concepts are distinct in general: a chief series is a maximal
562:
600:
575:
399:
25:
395:
313:
551:
Mathematical
Proceedings of the Cambridge Philosophical Society
426:
is a nontrivial normal subgroup properly contained in
112:
480:
is one of the elements (assuming a chief series for
433:, contradicting the definition of a chief series.
243:. Equivalently, there does not exist any subgroup
185:
598:
418:is cyclic and thus is generated by some integer
43:series, while a composition series is a maximal
508:have chief series. The converse also holds: if
457:of the chief factors and their multiplicities.
476:, one can always find a chief series in which
369:. In particular, a finite chief factor is a
557:(3). Cambridge University Press: 437–442.
365:, that is, they have no proper nontrivial
548:
361:. However, the chief factors are always
599:
574:
316:. That is, there may exist a subgroup
484:exists in the first place.) Also, if
422:, however the subgroup generated by 2
411:leads to an immediate contradiction:
542:
312:, chief factors are not necessarily
460:
13:
14:
618:
304:in a chief series are called the
1:
535:
436:
376:
373:of isomorphic simple groups.
55:
532:has a chief series as well.
381:
7:
226: − 1, is a
10:
623:
580:Algebra: A Graduate Course
468:Given any normal subgroup
87:is a finite collection of
563:10.1017/S0305004100055262
363:characteristically simple
367:characteristic subgroups
488:has a chief series and
228:minimal normal subgroup
308:of the series. Unlike
187:
443:Jordan–Hölder theorem
188:
282:may be added to it.
110:
64:under the action of
528:have chief series,
472: ⊆
455:isomorphism classes
310:composition factors
99: ⊆
66:inner automorphisms
35:It is similar to a
285:The factor groups
183:
37:composition series
576:Isaacs, I. Martin
453:, as well as the
357:is not normal in
614:
593:
567:
566:
546:
461:Other properties
192:
190:
189:
184:
173:
172:
154:
153:
141:
140:
128:
127:
89:normal subgroups
18:abstract algebra
622:
621:
617:
616:
615:
613:
612:
611:
607:Subgroup series
597:
596:
590:
582:. Brooks/Cole.
571:
570:
547:
543:
538:
463:
439:
432:
417:
410:
384:
379:
352:
338:
329:
303:
294:
273:
259:
242:
217:
208:
196:such that each
168:
164:
149:
145:
136:
132:
123:
119:
111:
108:
107:
98:
58:
12:
11:
5:
620:
610:
609:
595:
594:
588:
569:
568:
540:
539:
537:
534:
462:
459:
438:
435:
430:
415:
406:
383:
380:
378:
375:
371:direct product
347:
334:
324:
299:
289:
268:
255:
238:
213:
203:
198:quotient group
194:
193:
182:
179:
176:
171:
167:
163:
160:
157:
152:
148:
144:
139:
135:
131:
126:
122:
118:
115:
94:
71:In detail, if
57:
54:
9:
6:
4:
3:
2:
619:
608:
605:
604:
602:
591:
589:0-534-19002-2
585:
581:
577:
573:
572:
564:
560:
556:
552:
545:
541:
533:
531:
527:
523:
519:
515:
512:is normal in
511:
507:
503:
499:
495:
492:is normal in
491:
487:
483:
479:
475:
471:
466:
458:
456:
452:
449:of the group
448:
444:
434:
429:
425:
421:
414:
409:
405:
401:
397:
393:
389:
374:
372:
368:
364:
360:
356:
350:
346:
342:
337:
333:
327:
323:
319:
315:
311:
307:
306:chief factors
302:
298:
292:
288:
283:
281:
277:
271:
267:
263:
258:
254:
250:
246:
241:
237:
233:
229:
225:
221:
216:
212:
206:
202:
199:
180:
177:
174:
169:
165:
161:
158:
155:
150:
146:
142:
137:
133:
129:
124:
120:
116:
113:
106:
105:
104:
102:
97:
93:
90:
86:
82:
78:
74:
69:
67:
63:
53:
50:
48:
47:
42:
38:
33:
31:
27:
26:normal series
24:is a maximal
23:
19:
579:
554:
550:
544:
529:
525:
521:
517:
513:
509:
505:
501:
497:
496:, then both
493:
489:
485:
481:
477:
473:
469:
467:
464:
450:
440:
427:
423:
419:
412:
407:
403:
391:
387:
385:
358:
354:
348:
344:
340:
335:
331:
325:
321:
317:
305:
300:
296:
290:
286:
284:
279:
275:
269:
265:
261:
256:
252:
248:
244:
239:
235:
231:
223:
222:= 1, 2,...,
219:
214:
210:
204:
200:
195:
100:
95:
91:
84:
81:chief series
80:
72:
70:
61:
59:
51:
44:
40:
34:
22:chief series
21:
15:
536:References
437:Uniqueness
377:Properties
320:normal in
251:such that
247:normal in
56:Definition
516:and both
447:invariant
382:Existence
162:⊆
159:⋯
156:⊆
143:⊆
130:⊆
79:, then a
46:subnormal
601:Category
578:(1994).
274:for any
49:series.
400:abelian
586:
396:cyclic
353:, but
314:simple
218:, for
41:normal
28:for a
343:<
339:<
330:with
264:<
260:<
77:group
75:is a
30:group
584:ISBN
520:and
500:and
398:and
20:, a
559:doi
394:is
230:of
83:of
16:In
603::
555:84
553:.
351:+1
328:+1
293:+1
272:+1
207:+1
103:,
68:.
32:.
592:.
565:.
561::
530:G
526:N
524:/
522:G
518:N
514:G
510:N
506:N
504:/
502:G
498:N
494:G
490:N
486:G
482:G
478:N
474:G
470:N
451:G
431:1
428:N
424:a
420:a
416:1
413:N
408:i
404:N
392:Z
388:Z
359:G
355:A
349:i
345:N
341:A
336:i
332:N
326:i
322:N
318:A
301:i
297:N
295:/
291:i
287:N
280:G
276:i
270:i
266:N
262:A
257:i
253:N
249:G
245:A
240:i
236:N
234:/
232:G
224:n
220:i
215:i
211:N
209:/
205:i
201:N
181:,
178:G
175:=
170:n
166:N
151:2
147:N
138:1
134:N
125:0
121:N
117:=
114:1
101:G
96:i
92:N
85:G
73:G
62:G
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.