1233:
834:
Similarly, while being strictly characteristic (distinguished) is not transitive, it is true that every fully characteristic subgroup of a strictly characteristic subgroup is strictly characteristic.
434:
918:
Every subgroup that is fully characteristic is certainly strictly characteristic and characteristic; but a characteristic or even strictly characteristic subgroup need not be fully characteristic.
1018:
311:
and a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal. However, not every normal subgroup is characteristic. Here are several examples:
960:
1351:
1297:
1262:
1179:
1143:
1099:
722:
562:
2, all of which are normal. One of these is the cyclic subgroup, which is characteristic. The other two subgroups are dihedral; these are permuted by an
1188:
925:
is always a strictly characteristic subgroup, but it is not always fully characteristic. For example, the finite group of order 12,
355:, are both normal, but neither is characteristic. In particular, neither of these subgroups is invariant under the automorphism,
534:
of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup,
395:
806:
Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal.
666:
Every group has itself (the improper subgroup) and the trivial subgroup as two of its fully characteristic subgroups. The
982:
1531:
1506:
1481:
1445:
1473:
602:, surjectivity of an endomorphism implies injectivity, so a surjective endomorphism is an automorphism; thus being
1412:
930:
1469:
1043:
1553:
440:, every subgroup is normal; but every permutation of the 3 non-identity elements is an automorphism of
1327:
1273:
1238:
1155:
1119:
1075:
563:
328:
639:
1353:, is not contained in the center, so here the center is not a fully characteristic subgroup of
48:
1437:
59:; though the converse is not guaranteed. Examples of characteristic subgroups include the
8:
738:
667:
559:
60:
44:
1396:
52:
1527:
1502:
1477:
1441:
1400:
922:
389:
64:
1380:
1376:
531:
385:
24:
1103:
1047:
1035:
718:
710:
279:
267:
56:
548:
202:
1547:
1384:
437:
1107:
599:
595:
40:
28:
729:, the converse also holds: every fully characteristic subgroup is verbal.
20:
1027:
The relationship amongst these subgroup properties can be expressed as:
1461:
1264:
onto the indicated subgroup. Then the composition of the projection of
726:
714:
592:
16:
Subgroup mapped to itself under every automorphism of the parent group
1031:
36:
1228:{\displaystyle f:\mathbb {Z} _{2}<\rightarrow {\text{S}}_{3}}
737:
The property of being characteristic or fully characteristic is
1379:(or commutator subgroup) of a group is a verbal subgroup. The
538:, is characteristic, since it is the only subgroup of order 2.
446:, so the 3 subgroups of order 2 are not characteristic. Here
1522:
Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (2004).
566:
of the parent group, and are therefore not characteristic.
278:
that is invariant under all inner automorphisms is called
1102:(the group of order 12 that is the direct product of the
129:
It would be equivalent to require the stronger condition
713:, which is the image of a fully invariant subgroup of a
429:{\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}
670:
of a group is always a fully characteristic subgroup.
1330:
1276:
1241:
1191:
1158:
1122:
1078:
985:
933:
398:
1521:
1367:
Every subgroup of a cyclic group is characteristic.
610:. This is not the case anymore for infinite groups.
570:
1345:
1291:
1256:
1227:
1173:
1137:
1093:
1013:{\displaystyle 1\times \mathbb {Z} /2\mathbb {Z} }
1012:
954:
428:
1318:as its first factor, provides an endomorphism of
1545:
613:
1024:, which meets the center only in the identity.
1432:Dummit, David S.; Foote, Richard M. (2004).
1431:
955:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
717:under a homomorphism. More generally, any
1333:
1279:
1244:
1200:
1161:
1125:
1081:
1006:
993:
948:
935:
416:
401:
771:is a (fully) characteristic subgroup of
759:is a (fully) characteristic subgroup of
747:is a (fully) characteristic subgroup of
721:is always fully characteristic. For any
1546:
1496:
1403:is always a characteristic subgroup.
1390:
1324:under which the image of the center,
1042:⇐ Strictly characteristic subgroup ⇐
871:is not necessarily characteristic in
1460:
1370:
378:For a concrete example of this, let
1152:, contains subgroups isomorphic to
1116:is isomorphic to its second factor
618:For an even stronger constraint, a
256:
180:
55:, every characteristic subgroup is
13:
704:
261:
39:that is mapped to itself by every
14:
1565:
1058:
1362:
1346:{\displaystyle \mathbb {Z} _{2}}
1292:{\displaystyle \mathbb {Z} _{2}}
1257:{\displaystyle \mathbb {Z} _{2}}
1174:{\displaystyle \mathbb {Z} _{2}}
1138:{\displaystyle \mathbb {Z} _{2}}
1094:{\displaystyle \mathbb {Z} _{2}}
579:strictly characteristic subgroup
571:Strictly characteristic subgroup
375:, that switches the two factors.
1413:Characteristically simple group
1387:is a fully invariant subgroup.
1305:, followed by the inclusion of
913:
732:
709:An even stronger constraint is
478:and consider the automorphism,
321:be a nontrivial group, and let
282:; also, an invariant subgroup.
201:induces an automorphism of the
1515:
1490:
1454:
1425:
1145:. Note that the first factor,
837:However, unlike normality, if
725:, and, in particular, for any
210:, which yields a homomorphism
166:implies the reverse inclusion
23:, particularly in the area of
1:
1470:Graduate Texts in Mathematics
1418:
1044:Fully characteristic subgroup
620:fully characteristic subgroup
614:Fully characteristic subgroup
70:
963:, has a homomorphism taking
642:under every endomorphism of
7:
1406:
1110:of order 2). The center of
1053:
679:induces an endomorphism of
626:; cf. invariant subgroup),
591:, which is invariant under
10:
1570:
1524:Combinatorial Group Theory
979:, which takes the center,
265:
91:if for every automorphism
1526:. Dover. pp. 74–85.
1501:. Dover. pp. 45–46.
1235:be the morphism mapping
624:fully invariant subgroup
436:). Since this group is
195:, every automorphism of
1270:onto its second factor
1040:Characteristic subgroup
638:, is a group remaining
604:strictly characteristic
392:to the direct product,
341:. Then the subgroups,
241:of a given index, then
143:for every automorphism
89:characteristic subgroup
33:characteristic subgroup
1347:
1293:
1258:
1229:
1175:
1139:
1095:
1014:
956:
673:Every endomorphism of
587:distinguished subgroup
430:
235:has a unique subgroup
1438:John Wiley & Sons
1348:
1294:
1259:
1230:
1176:
1140:
1096:
1020:, into a subgroup of
1015:
957:
685:, which yields a map
431:
247:is characteristic in
1497:Scott, W.R. (1987).
1328:
1274:
1239:
1189:
1156:
1120:
1076:
983:
931:
521:is not contained in
396:
1554:Subgroup properties
1063:Consider the group
668:commutator subgroup
558:has 3 subgroups of
61:commutator subgroup
1397:identity component
1391:Topological groups
1343:
1289:
1254:
1225:
1171:
1135:
1091:
1010:
952:
865:, then in general
723:reduced free group
564:outer automorphism
426:
53:inner automorphism
1401:topological group
1371:Subgroup functors
1217:
1106:of order 6 and a
923:center of a group
853:is a subgroup of
606:is equivalent to
65:center of a group
47:. Because every
1561:
1538:
1537:
1519:
1513:
1512:
1494:
1488:
1487:
1458:
1452:
1451:
1436:(3rd ed.).
1434:Abstract Algebra
1429:
1381:torsion subgroup
1377:derived subgroup
1358:
1352:
1350:
1349:
1344:
1342:
1341:
1336:
1323:
1317:
1311:
1304:
1298:
1296:
1295:
1290:
1288:
1287:
1282:
1269:
1263:
1261:
1260:
1255:
1253:
1252:
1247:
1234:
1232:
1231:
1226:
1224:
1223:
1218:
1215:
1209:
1208:
1203:
1184:
1180:
1178:
1177:
1172:
1170:
1169:
1164:
1151:
1144:
1142:
1141:
1136:
1134:
1133:
1128:
1115:
1101:
1100:
1098:
1097:
1092:
1090:
1089:
1084:
1023:
1019:
1017:
1016:
1011:
1009:
1001:
996:
978:
974:
962:
961:
959:
958:
953:
951:
943:
938:
909:
876:
870:
864:
858:
852:
846:
830:
801:
776:
770:
764:
758:
752:
746:
700:
684:
678:
661:
647:
637:
631:
589:
588:
581:
580:
557:
546:
537:
532:quaternion group
526:
520:
512:
477:
465:
445:
435:
433:
432:
427:
425:
424:
419:
410:
409:
404:
386:Klein four-group
383:
374:
354:
347:
340:
326:
320:
310:
295:
277:
257:Related concepts
252:
246:
240:
234:
225:
209:
200:
194:
181:Basic properties
176:
165:
154:
148:
142:
136:
124:
113:
102:
96:
86:
80:
25:abstract algebra
1569:
1568:
1564:
1563:
1562:
1560:
1559:
1558:
1544:
1543:
1542:
1541:
1534:
1520:
1516:
1509:
1495:
1491:
1484:
1459:
1455:
1448:
1430:
1426:
1421:
1409:
1393:
1373:
1365:
1354:
1337:
1332:
1331:
1329:
1326:
1325:
1319:
1313:
1310:
1306:
1300:
1283:
1278:
1277:
1275:
1272:
1271:
1265:
1248:
1243:
1242:
1240:
1237:
1236:
1219:
1214:
1213:
1204:
1199:
1198:
1190:
1187:
1186:
1182:
1181:, for instance
1165:
1160:
1159:
1157:
1154:
1153:
1150:
1146:
1129:
1124:
1123:
1121:
1118:
1117:
1111:
1104:symmetric group
1085:
1080:
1079:
1077:
1074:
1073:
1071:
1064:
1061:
1056:
1048:Verbal subgroup
1036:Normal subgroup
1021:
1005:
997:
992:
984:
981:
980:
976:
964:
947:
939:
934:
932:
929:
928:
926:
916:
881:
872:
866:
860:
854:
848:
838:
810:
781:
772:
766:
760:
754:
748:
742:
735:
719:verbal subgroup
711:verbal subgroup
707:
705:Verbal subgroup
686:
680:
674:
652:
643:
633:
627:
616:
586:
585:
578:
577:
573:
552:
542:
535:
522:
514:
479:
467:
447:
441:
420:
415:
414:
405:
400:
399:
397:
394:
393:
379:
356:
349:
342:
332:
322:
316:
300:
286:
273:
270:
268:Normal subgroup
264:
262:Normal subgroup
259:
248:
242:
236:
230:
211:
205:
196:
186:
183:
167:
156:
150:
144:
138:
130:
116:
104:
98:
92:
82:
76:
73:
49:conjugation map
17:
12:
11:
5:
1567:
1557:
1556:
1540:
1539:
1532:
1514:
1507:
1489:
1482:
1453:
1446:
1423:
1422:
1420:
1417:
1416:
1415:
1408:
1405:
1392:
1389:
1372:
1369:
1364:
1361:
1340:
1335:
1308:
1299:, followed by
1286:
1281:
1251:
1246:
1222:
1212:
1207:
1202:
1197:
1194:
1168:
1163:
1148:
1132:
1127:
1088:
1083:
1069:
1060:
1059:Finite example
1057:
1055:
1052:
1051:
1050:
1008:
1004:
1000:
995:
991:
988:
950:
946:
942:
937:
915:
912:
911:
910:
832:
831:
804:
803:
734:
731:
706:
703:
664:
663:
615:
612:
608:characteristic
572:
569:
568:
567:
549:dihedral group
539:
528:
423:
418:
413:
408:
403:
376:
329:direct product
297:
296:
272:A subgroup of
266:Main article:
263:
260:
258:
255:
203:quotient group
182:
179:
72:
69:
43:of the parent
15:
9:
6:
4:
3:
2:
1566:
1555:
1552:
1551:
1549:
1535:
1533:0-486-43830-9
1529:
1525:
1518:
1510:
1508:0-486-65377-3
1504:
1500:
1493:
1485:
1483:0-387-95385-X
1479:
1475:
1471:
1467:
1463:
1457:
1449:
1447:0-471-43334-9
1443:
1439:
1435:
1428:
1424:
1414:
1411:
1410:
1404:
1402:
1398:
1388:
1386:
1385:abelian group
1382:
1378:
1368:
1363:Cyclic groups
1360:
1357:
1338:
1322:
1316:
1303:
1284:
1268:
1249:
1220:
1211:<→
1210:
1205:
1195:
1192:
1166:
1130:
1114:
1109:
1105:
1086:
1067:
1049:
1045:
1041:
1037:
1033:
1030:
1029:
1028:
1025:
1002:
998:
989:
986:
972:
968:
944:
940:
924:
919:
908:
904:
900:
896:
892:
888:
884:
880:
879:
878:
875:
869:
863:
857:
851:
845:
841:
835:
829:
825:
821:
817:
813:
809:
808:
807:
800:
796:
792:
788:
784:
780:
779:
778:
775:
769:
763:
757:
751:
745:
740:
730:
728:
724:
720:
716:
712:
702:
698:
694:
690:
683:
677:
671:
669:
660:
656:
651:
650:
649:
646:
641:
636:
632:, of a group
630:
625:
621:
611:
609:
605:
601:
600:finite groups
597:
596:endomorphisms
594:
590:
582:
565:
561:
556:
550:
547:is even, the
545:
540:
533:
529:
525:
518:
511:
507:
503:
499:
495:
491:
487:
483:
475:
471:
463:
459:
455:
451:
444:
439:
421:
411:
406:
391:
387:
382:
377:
372:
368:
364:
360:
352:
346:
339:
335:
330:
325:
319:
314:
313:
312:
308:
304:
294:
290:
285:
284:
283:
281:
276:
269:
254:
251:
245:
239:
233:
227:
223:
219:
215:
208:
204:
199:
193:
189:
178:
174:
170:
164:
160:
153:
147:
141:
134:
127:
125:
123:
119:
114:; then write
112:
108:
101:
95:
90:
85:
79:
68:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
1523:
1517:
1499:Group Theory
1498:
1492:
1465:
1456:
1433:
1427:
1394:
1374:
1366:
1355:
1320:
1314:
1301:
1266:
1112:
1108:cyclic group
1065:
1062:
1039:
1026:
970:
966:
920:
917:
914:Containments
906:
902:
898:
894:
890:
886:
882:
873:
867:
861:
855:
849:
843:
839:
836:
833:
827:
823:
819:
815:
811:
805:
798:
794:
790:
786:
782:
773:
767:
761:
755:
749:
743:
736:
733:Transitivity
708:
696:
692:
688:
681:
675:
672:
665:
658:
654:
644:
634:
628:
623:
619:
617:
607:
603:
584:
576:
574:
554:
543:
523:
516:
509:
505:
501:
497:
493:
489:
485:
481:
473:
469:
461:
457:
453:
449:
442:
380:
370:
366:
362:
358:
350:
344:
337:
333:
323:
317:
306:
302:
298:
292:
288:
274:
271:
249:
243:
237:
231:
228:
221:
217:
213:
206:
197:
191:
187:
184:
172:
168:
162:
158:
151:
145:
139:
132:
128:
121:
117:
115:
110:
106:
99:
93:
88:
87:is called a
83:
77:
74:
41:automorphism
32:
29:group theory
18:
1462:Lang, Serge
977:((1, 2), 0)
859:containing
648:; that is,
466:. Consider
81:of a group
75:A subgroup
21:mathematics
1419:References
1183:{e, (12)}
1022:Sym(3) × 1
739:transitive
727:free group
715:free group
593:surjective
390:isomorphic
388:(which is
155:, because
103:, one has
71:Definition
990:×
927:Sym(3) ×
653:∀φ ∈ End(
640:invariant
551:of order
412:×
287:∀φ ∈ Inn(
27:known as
1548:Category
1474:Springer
1464:(2002).
1407:See also
1054:Examples
1032:Subgroup
691:) → End(
305:) ⊆ Aut(
216:) → Aut(
63:and the
37:subgroup
1466:Algebra
765:, then
657:)： φ ≤
622:(also,
583:, or a
536:{1, −1}
530:In the
513:; then
438:abelian
384:be the
327:be the
291:)： φ ≤
1530:
1505:
1480:
1444:
1383:of an
1185:; let
753:, and
598:. For
343:{1} ×
299:Since
280:normal
185:Given
57:normal
51:is an
1399:of a
1312:into
905:char
897:<
893:<
885:char
842:char
814:char
797:char
789:char
785:char
741:; if
560:index
468:H = {
448:V = {
365:) → (
353:× {1}
190:char
120:char
45:group
35:is a
1528:ISBN
1503:ISBN
1478:ISBN
1442:ISBN
1395:The
1375:The
921:The
847:and
687:End(
508:) =
504:, T(
500:) =
496:, T(
492:) =
488:, T(
484:) =
348:and
315:Let
301:Inn(
212:Aut(
171:≤ φ(
161:) ≤
109:) ≤
31:, a
1068:= S
975:to
682:G/H
541:If
229:If
207:G/H
149:of
97:of
19:In
1550::
1476:.
1472:.
1468:.
1440:.
1359:.
1072:×
1046:⇐
1038:⇐
1034:⇐
969:,
901:⇏
889:,
877:.
826:⊲
822:⇒
818:⊲
793:⇒
777:.
701:.
575:A
515:T(
510:ab
506:ab
480:T(
472:,
464:}
462:ab
460:,
456:,
452:,
369:,
361:,
336:×
331:,
253:.
226:.
177:.
157:φ(
137:=
131:φ(
126:.
105:φ(
67:.
1536:.
1511:.
1486:.
1450:.
1356:G
1339:2
1334:Z
1321:G
1315:G
1309:3
1307:S
1302:f
1285:2
1280:Z
1267:G
1250:2
1245:Z
1221:3
1216:S
1206:2
1201:Z
1196::
1193:f
1167:2
1162:Z
1149:3
1147:S
1131:2
1126:Z
1113:G
1087:2
1082:Z
1070:3
1066:G
1007:Z
1003:2
999:/
994:Z
987:1
973:)
971:y
967:π
965:(
949:Z
945:2
941:/
936:Z
907:K
903:H
899:G
895:K
891:H
887:G
883:H
874:K
868:H
862:H
856:G
850:K
844:G
840:H
828:G
824:H
820:G
816:K
812:H
802:.
799:G
795:H
791:G
787:K
783:H
774:G
768:H
762:G
756:K
750:K
744:H
699:)
697:H
695:/
693:G
689:G
676:G
662:.
659:H
655:G
645:G
635:G
629:H
555:n
553:2
544:n
527:.
524:H
519:)
517:H
502:a
498:b
494:b
490:a
486:e
482:e
476:}
474:a
470:e
458:b
454:a
450:e
443:V
422:2
417:Z
407:2
402:Z
381:V
373:)
371:x
367:y
363:y
359:x
357:(
351:H
345:H
338:H
334:H
324:G
318:H
309:)
307:G
303:G
293:H
289:G
275:H
250:G
244:H
238:H
232:G
224:)
222:H
220:/
218:G
214:G
198:G
192:G
188:H
175:)
173:H
169:H
163:H
159:H
152:G
146:φ
140:H
135:)
133:H
122:G
118:H
111:H
107:H
100:G
94:φ
84:G
78:H
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