206:
289:
256:
352:
332:
309:
229:
119:
99:
79:
510:
488:
480:
377:
402:, that is, a subnormal subgroup of a subnormal subgroup is subnormal. The relation of subnormality can be defined as the
127:
528:
362:
434:
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472:
373:
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32:
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47:
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214:
104:
84:
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21:
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17:
502:
497:
Ballester-Bolinches, Adolfo; Esteban-Romero, Ramon; Asaad, Mohamed (2010),
25:
358:
A 1-subnormal subgroup is a proper normal subgroup (and vice versa).
46:
if there is a finite chain of subgroups of the group, each one
394:
Every 2-subnormal subgroup is a conjugate-permutable subgroup.
340:
320:
297:
264:
237:
217:
130:
107:
87:
67:
387:
that is also subnormal, is normal. In particular, a
201:{\displaystyle H=H_{0},H_{1},H_{2},\ldots ,H_{k}=G}
369:if and only if each of its subgroups is subnormal.
346:
326:
303:
283:
250:
223:
200:
113:
93:
73:
8:
314:A subnormal subgroup is a subgroup that is
391:is subnormal if and only if it is normal.
339:
319:
296:
269:
263:
242:
236:
216:
186:
167:
154:
141:
129:
106:
86:
66:
354:. Some facts about subnormal subgroups:
334:-subnormal for some positive integer
7:
14:
380:, of a finite group is subnormal.
477:A Course in the Theory of Groups
398:The property of subnormality is
409:If every subnormal subgroup of
406:of the relation of normality.
1:
378:conjugate-permutable subgroup
376:, and, more generally, every
545:
50:in the next, beginning at
499:Products of Finite Groups
363:finitely generated group
435:Characteristic subgroup
284:{\displaystyle H_{i+1}}
121:if there are subgroups
348:
328:
305:
285:
252:
225:
202:
115:
95:
75:
349:
329:
306:
286:
253:
251:{\displaystyle H_{i}}
226:
203:
116:
96:
76:
479:, Berlin, New York:
473:Robinson, Derek J.S.
374:quasinormal subgroup
338:
318:
295:
262:
235:
215:
128:
105:
85:
65:
529:Subgroup properties
455:Descendant subgroup
450:Ascendant subgroup
404:transitive closure
385:pronormal subgroup
344:
324:
301:
281:
248:
221:
198:
111:
91:
71:
40:subnormal subgroup
20:, in the field of
512:978-3-11-022061-2
503:Walter de Gruyter
490:978-0-387-94461-6
347:{\displaystyle k}
327:{\displaystyle k}
304:{\displaystyle i}
224:{\displaystyle G}
114:{\displaystyle G}
94:{\displaystyle k}
74:{\displaystyle H}
536:
515:
493:
353:
351:
350:
345:
333:
331:
330:
325:
310:
308:
307:
302:
290:
288:
287:
282:
280:
279:
257:
255:
254:
249:
247:
246:
230:
228:
227:
222:
207:
205:
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199:
191:
190:
172:
171:
159:
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146:
145:
120:
118:
117:
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100:
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80:
78:
77:
72:
544:
543:
539:
538:
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534:
533:
519:
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513:
496:
491:
481:Springer-Verlag
471:
468:
460:Serial subgroup
431:
336:
335:
316:
315:
293:
292:
265:
260:
259:
238:
233:
232:
213:
212:
182:
163:
150:
137:
126:
125:
103:
102:
83:
82:
63:
62:
12:
11:
5:
542:
540:
532:
531:
521:
520:
517:
516:
511:
494:
489:
467:
464:
463:
462:
457:
452:
447:
445:Normal closure
442:
437:
430:
427:
396:
395:
392:
389:Sylow subgroup
381:
370:
359:
343:
323:
300:
278:
275:
272:
268:
245:
241:
220:
209:
208:
197:
194:
189:
185:
181:
178:
175:
170:
166:
162:
157:
153:
149:
144:
140:
136:
133:
110:
101:-subnormal in
90:
70:
54:and ending at
13:
10:
9:
6:
4:
3:
2:
541:
530:
527:
526:
524:
514:
508:
504:
500:
495:
492:
486:
482:
478:
474:
470:
469:
465:
461:
458:
456:
453:
451:
448:
446:
443:
441:
438:
436:
433:
432:
428:
426:
424:
420:
416:
413:is normal in
412:
407:
405:
401:
393:
390:
386:
382:
379:
375:
371:
368:
364:
360:
357:
356:
355:
341:
321:
312:
298:
276:
273:
270:
266:
258:is normal in
243:
239:
218:
195:
192:
187:
183:
179:
176:
173:
168:
164:
160:
155:
151:
147:
142:
138:
134:
131:
124:
123:
122:
108:
88:
68:
61:In notation,
59:
57:
53:
49:
45:
41:
37:
34:
30:
27:
23:
19:
498:
476:
421:is called a
418:
414:
410:
408:
397:
313:
210:
60:
55:
51:
43:
39:
35:
28:
22:group theory
15:
440:Normal core
31:of a given
18:mathematics
466:References
400:transitive
231:such that
367:nilpotent
291:for each
177:…
523:Category
475:(1996),
429:See also
26:subgroup
423:T-group
417:, then
509:
487:
383:Every
372:Every
48:normal
38:is a
33:group
507:ISBN
485:ISBN
24:, a
365:is
211:of
81:is
42:of
16:In
525::
505:,
501:,
483:,
425:.
361:A
311:.
58:.
419:G
415:G
411:G
342:k
322:k
299:i
277:1
274:+
271:i
267:H
244:i
240:H
219:G
196:G
193:=
188:k
184:H
180:,
174:,
169:2
165:H
161:,
156:1
152:H
148:,
143:0
139:H
135:=
132:H
109:G
89:k
69:H
56:G
52:H
44:G
36:G
29:H
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