484:
1448:
711:
1236:
330:
2858:
2782:
3643:
Centralizers in rings and in algebras over a field are subrings and subalgebras over a field, respectively; centralizers in Lie rings and in Lie algebras are Lie subrings and Lie subalgebras, respectively.
1845:
The normalizer of H with respect to the group G are all elements of G that yield the set H (potentially permuted) when the group operation is applied. Working out the example for each element of G:
1551:
1011:
113:
207:
1350:
821:
1753:
1842:
Note that is the identity permutation in G and retains the order of each element and is the permutation that fixes the first element and swaps the second and third element.
585:
1511:
1483:
1342:
1138:
1130:
1094:
237:
2540:
2423:
2306:
2189:
1837:
790:
2807:
2664:
2072:
1958:
866:
2612:
2704:
2684:
1031:
841:
754:
734:
149:
2463:
2346:
2229:
2112:
1998:
1884:
3763:
40:
479:{\displaystyle \mathrm {C} _{G}(S)=\left\{g\in G\mid gs=sg{\text{ for all }}s\in S\right\}=\left\{g\in G\mid gsg^{-1}=s{\text{ for all }}s\in S\right\},}
3964:
3929:
3833:
3802:
948:. The same notational conventions mentioned above for centralizers also apply to normalizers. The normalizer should not be confused with the
3821:
The Lie Theory of
Connected Pro-Lie Groups: A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups
2624:
The centralizer of the group G is the set of elements that leave each element of H unchanged. It's clear that the only such element in S
2709:
3988:
2617:
A group is considered simple if the normalizer with respect to a subset is always the identity and itself. Here, it's clear that S
3909:
3913:
3825:
276:
is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring
1516:
3753:
3678:
3506:
251:. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets
128:
4014:
3974:
3950:
3903:
489:
where only the first definition applies to semigroups. If there is no ambiguity about the group in question, the
958:
79:
3794:
1453:
While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the
1241:
The definition of centralizers for Lie rings is linked to the definition for rings in the following way. If
502:
1443:{\displaystyle \mathrm {N} _{\mathfrak {L}}(S)=\{x\in {\mathfrak {L}}\mid \in S{\text{ for all }}s\in S\}.}
299:
in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.
706:{\displaystyle \mathrm {N} _{G}(S)=\left\{g\in G\mid gS=Sg\right\}=\left\{g\in G\mid gSg^{-1}=S\right\},}
174:
4029:
4019:
1231:{\displaystyle \mathrm {C} _{\mathfrak {L}}(S)=\{x\in {\mathfrak {L}}\mid =0{\text{ for all }}s\in S\}.}
530:
1575:
4024:
1492:
1464:
1323:
1111:
1075:
1046:
2816:
3768:
1558:
795:
70:
33:
216:
3980:
3956:
3437:
3426:
272:
3984:
3960:
3925:
3829:
3798:
3359:
2468:
2351:
2234:
2117:
1765:
949:
759:
168:
3819:
3788:
2003:
1889:
3917:
3998:
3939:
2549:
3994:
3946:
3935:
2922:
2787:
2644:
846:
2689:
2669:
1016:
826:
739:
719:
134:
2430:
2313:
2196:
2079:
1965:
1851:
4008:
3223:
120:
3367:
3335:
2810:
51:
533:. With this latter notation, one must be careful to avoid confusion between the
3790:
Advanced Topics in Linear
Algebra: Weaving Matrix Problems Through the Weyr Form
2861:
1097:
289:
266:
47:
32:"Normalizer" redirects here. For the process of increasing audio amplitude, see
3748:
2978:
1250:
3758:
2982:
1553:
is the largest Lie subring (or Lie subalgebra, as the case may be) in which
1454:
296:
259:
3912:, vol. 100 (reprint of the 1994 original ed.), Providence, RI:
1101:
244:
17:
716:
where again only the first definition applies to semigroups. If the set
281:
3921:
63:
39:"Centralizer" redirects here. For centralizers of Banach spaces, see
2777:{\displaystyle S'=\{x\in A\mid sx=xs{\text{ for every }}s\in S\}.}
288:. This article also deals with centralizers and normalizers in a
3858:
3856:
3854:
2542:; therefore is not in the Normalizer(H) with respect to G.
2425:; therefore is not in the Normalizer(H) with respect to G.
2308:; therefore is not in the Normalizer(H) with respect to G.
2191:; therefore is not in the Normalizer(H) with respect to G.
239:
fixed under conjugation. The centralizer and normalizer of
525:}). Another less common notation for the centralizer is Z(
3787:
2074:; therefore is in the Normalizer(H) with respect to G.
1960:; therefore is in the Normalizer(H) with respect to G.
3151:
commute with each other, then the largest subgroup of
1104:) with Lie product , then the centralizer of a subset
27:
Special types of subgroups encountered in group theory
2819:
2790:
2712:
2692:
2672:
2647:
2552:
2471:
2433:
2354:
2316:
2237:
2199:
2120:
2082:
2006:
1968:
1892:
1854:
1768:
1578:
1519:
1495:
1467:
1353:
1326:
1141:
1114:
1078:
1037:
Ring, algebra over a field, Lie ring, and Lie algebra
1019:
961:
849:
829:
798:
762:
742:
722:
588:
333:
219:
213:
that satisfy the weaker condition of leaving the set
177:
137:
82:
2614:
since both these group elements preserve the set H.
1755:(the symmetric group of permutations of 3 elements).
258:Suitably formulated, the definitions also apply to
2973:) acts by conjugation as a group of bijections on
2852:
2801:
2776:
2698:
2678:
2658:
2606:
2546:Therefore, the Normalizer(H) with respect to G is
2534:
2457:
2417:
2340:
2300:
2223:
2183:
2106:
2066:
1992:
1952:
1878:
1831:
1747:
1545:
1505:
1477:
1442:
1336:
1230:
1124:
1088:
1025:
1005:
860:
835:
815:
784:
748:
728:
705:
478:
231:
201:
143:
107:
3045:it is a central tool in the theory of Lie groups.
3818:Karl Heinrich Hofmann; Sidney A. Morris (2007).
3979:(republication of the 1962 original ed.),
3027:, and especially if the torus is maximal (i.e.
1546:{\displaystyle \mathrm {N} _{\mathfrak {L}}(S)}
8:
3764:Multipliers and centralizers (Banach spaces)
3681:deals with situations where equality occurs.
2768:
2724:
2601:
2553:
2520:
2472:
2403:
2355:
2286:
2238:
2169:
2121:
2055:
2007:
1941:
1893:
1823:
1775:
1742:
1598:
1434:
1380:
1222:
1168:
41:Multipliers and centralizers (Banach spaces)
3650:in a Lie ring contains the centralizer of
1006:{\displaystyle C_{G}(S)\subseteq N_{G}(S)}
932:. That is, elements of the centralizer of
493:can be suppressed from the notation. When
2818:
2789:
2754:
2711:
2691:
2671:
2646:
2551:
2470:
2432:
2353:
2315:
2236:
2198:
2119:
2081:
2005:
1967:
1891:
1853:
1767:
1589:
1577:
1527:
1526:
1521:
1518:
1497:
1496:
1494:
1469:
1468:
1466:
1420:
1390:
1389:
1361:
1360:
1355:
1352:
1328:
1327:
1325:
1208:
1178:
1177:
1149:
1148:
1143:
1140:
1116:
1115:
1113:
1080:
1079:
1077:
1018:
988:
966:
960:
848:
828:
797:
767:
761:
741:
721:
680:
595:
590:
587:
454:
439:
390:
340:
335:
332:
218:
184:
179:
176:
136:
108:{\displaystyle \operatorname {C} _{G}(S)}
87:
81:
3874:
3862:
2935:), being the kernel of the homomorphism
2847:′′′′′
529:), which parallels the notation for the
3779:
1061:is exactly as defined for groups, with
3886:
3688:is an additive subgroup of a Lie ring
7:
3409:, the N/C theorem also implies that
940:, but elements of the normalizer of
1528:
1498:
1470:
1391:
1362:
1329:
1179:
1150:
1117:
1081:
202:{\displaystyle \mathrm {N} _{G}(S)}
3677:but is not necessarily equal. The
3082:. Containment occurs exactly when
2876:The centralizer and normalizer of
1522:
1356:
1144:
876:are similar but not identical. If
591:
336:
180:
84:
25:
1759:Take a subset H of the group G:
1748:{\displaystyle G=S_{3}=\{,,,,,\}}
3702:) is the largest Lie subring of
3910:Graduate Studies in Mathematics
3717:is a Lie subring of a Lie ring
3635:Rings and algebras over a field
3122:, then the largest subgroup of
1506:{\displaystyle {\mathfrak {L}}}
1478:{\displaystyle {\mathfrak {L}}}
1337:{\displaystyle {\mathfrak {L}}}
1320:of a Lie algebra (or Lie ring)
1125:{\displaystyle {\mathfrak {L}}}
1089:{\displaystyle {\mathfrak {L}}}
2860:; i.e. a commutant is its own
2853:{\displaystyle S'=S'''=S'''''}
2598:
2580:
2574:
2556:
2517:
2499:
2493:
2475:
2452:
2434:
2400:
2382:
2376:
2358:
2335:
2317:
2283:
2265:
2259:
2241:
2218:
2200:
2166:
2148:
2142:
2124:
2101:
2083:
2052:
2034:
2028:
2010:
1987:
1969:
1938:
1920:
1914:
1896:
1873:
1855:
1820:
1802:
1796:
1778:
1739:
1721:
1715:
1697:
1691:
1673:
1667:
1649:
1643:
1625:
1619:
1601:
1540:
1534:
1411:
1399:
1374:
1368:
1199:
1187:
1162:
1156:
1000:
994:
978:
972:
779:
773:
607:
601:
352:
346:
196:
190:
102:
96:
1:
3914:American Mathematical Society
3826:European Mathematical Society
1281:with the bracket product as L
1245:is an associative ring, then
816:{\displaystyle G'\subseteq G}
271:centralizer of a subset of a
127:, or equivalently, such that
3955:, vol. 1 (2 ed.),
3539:)), and the subgroup of Inn(
936:must commute pointwise with
575:in the group (or semigroup)
232:{\displaystyle S\subseteq G}
3130:is normal is the subgroup N
1489:is an additive subgroup of
1316:The normalizer of a subset
906:is in the normalizer, then
4046:
3905:Algebra: a graduate course
3902:Isaacs, I. Martin (2009),
3754:Double centralizer theorem
3679:double centralizer theorem
3147:such that all elements of
2666:denote the centralizer of
2628:is the identity element .
1057:, then the centralizer of
1013:and both are subgroups of
38:
31:
3973:Jacobson, Nathan (1979),
3180:self-normalizing subgroup
880:is in the centralizer of
3485:, then we can describe N
2535:{\displaystyle \{,\}!=H}
2465:when applied to H =>
2418:{\displaystyle \{,\}!=H}
2348:when applied to H =>
2301:{\displaystyle \{,\}!=H}
2231:when applied to H =>
2184:{\displaystyle \{,\}!=H}
2114:when applied to H =>
2000:when applied to H =>
1886:when applied to H =>
1832:{\displaystyle H=\{,\}.}
928:possibly different from
843:is a normal subgroup of
792:is the largest subgroup
785:{\displaystyle N_{G}(S)}
320:of group (or semigroup)
3795:Oxford University Press
3607:. If so, then in fact,
3421:), the subgroup of Aut(
3417:) is isomorphic to Inn(
2621:is not a simple group.
2067:{\displaystyle \{,\}=H}
1953:{\displaystyle \{,\}=H}
1277:. If we denote the set
944:need only commute with
892:, then it must be that
151:leaves each element of
3848:Jacobson (2009), p. 41
3155:whose center contains
2880:are both subgroups of
2854:
2803:
2778:
2700:
2680:
2660:
2608:
2536:
2459:
2419:
2342:
2302:
2225:
2185:
2108:
2068:
1994:
1954:
1880:
1833:
1749:
1547:
1507:
1479:
1444:
1338:
1232:
1126:
1090:
1027:
1007:
862:
837:
817:
786:
756:, then the normalizer
750:
730:
707:
480:
233:
203:
145:
123:with every element of
109:
3362:to a subgroup of Aut(
2855:
2804:
2779:
2756: for every
2701:
2681:
2661:
2609:
2607:{\displaystyle \{,\}}
2537:
2460:
2420:
2343:
2303:
2226:
2186:
2109:
2069:
1995:
1955:
1881:
1834:
1750:
1548:
1508:
1480:
1445:
1339:
1233:
1127:
1091:
1028:
1008:
868:. The definitions of
863:
838:
818:
787:
751:
731:
708:
481:
234:
204:
146:
110:
3517:: the stabilizer of
3425:) consisting of all
3246:For singleton sets,
2817:
2788:
2710:
2690:
2670:
2645:
2550:
2469:
2431:
2352:
2314:
2235:
2197:
2118:
2080:
2004:
1966:
1890:
1852:
1766:
1576:
1569:Consider the group
1517:
1493:
1465:
1351:
1324:
1301:Lie ring centralizer
1139:
1112:
1076:
1047:algebra over a field
1017:
959:
847:
827:
796:
760:
740:
720:
586:
331:
217:
175:
135:
80:
3769:Stabilizer subgroup
3427:inner automorphisms
3281:are two subsets of
3078:) need not contain
1422: for all
1287:, then clearly the
1210: for all
456: for all
392: for all
308:Group and semigroup
34:Audio normalization
3981:Dover Publications
3957:Dover Publications
3646:The normalizer of
3505:) in terms of the
3438:group homomorphism
2850:
2802:{\displaystyle S'}
2799:
2774:
2696:
2676:
2659:{\displaystyle S'}
2656:
2604:
2532:
2455:
2415:
2338:
2298:
2221:
2181:
2104:
2064:
1990:
1950:
1876:
1829:
1745:
1543:
1503:
1475:
1440:
1334:
1228:
1122:
1086:
1023:
1003:
861:{\displaystyle G'}
858:
833:
813:
782:
746:
726:
703:
476:
229:
199:
141:
105:
3966:978-0-486-47189-1
3931:978-0-8218-4799-2
3835:978-3-03719-032-6
3804:978-0-19-979373-0
3159:is the subgroup C
3118:is a subgroup of
3093:is a subgroup of
2757:
2699:{\displaystyle A}
2686:in the semigroup
2679:{\displaystyle S}
1423:
1263:. Of course then
1249:can be given the
1211:
1132:is defined to be
1026:{\displaystyle G}
836:{\displaystyle S}
749:{\displaystyle G}
736:is a subgroup of
729:{\displaystyle S}
457:
393:
144:{\displaystyle g}
16:(Redirected from
4037:
4015:Abstract algebra
4001:
3969:
3947:Jacobson, Nathan
3942:
3890:
3884:
3878:
3872:
3866:
3860:
3849:
3846:
3840:
3839:
3815:
3809:
3808:
3784:
3737:
3629:
3606:
3597:for some subset
3596:
3578:self-bicommutant
3484:
3453:
3408:
3390:
3366:), the group of
3334:states that the
3318:
3301:
3273:By symmetry, if
3269:
3242:
3204:
3182:
3181:
3044:
3026:
2952:
2910:
2859:
2857:
2856:
2851:
2849:
2838:
2827:
2808:
2806:
2805:
2800:
2798:
2783:
2781:
2780:
2775:
2758:
2755:
2720:
2705:
2703:
2702:
2697:
2685:
2683:
2682:
2677:
2665:
2663:
2662:
2657:
2655:
2613:
2611:
2610:
2605:
2541:
2539:
2538:
2533:
2464:
2462:
2461:
2458:{\displaystyle }
2456:
2424:
2422:
2421:
2416:
2347:
2345:
2344:
2341:{\displaystyle }
2339:
2307:
2305:
2304:
2299:
2230:
2228:
2227:
2224:{\displaystyle }
2222:
2190:
2188:
2187:
2182:
2113:
2111:
2110:
2107:{\displaystyle }
2105:
2073:
2071:
2070:
2065:
1999:
1997:
1996:
1993:{\displaystyle }
1991:
1959:
1957:
1956:
1951:
1885:
1883:
1882:
1879:{\displaystyle }
1877:
1838:
1836:
1835:
1830:
1754:
1752:
1751:
1746:
1594:
1593:
1552:
1550:
1549:
1544:
1533:
1532:
1531:
1525:
1512:
1510:
1509:
1504:
1502:
1501:
1484:
1482:
1481:
1476:
1474:
1473:
1449:
1447:
1446:
1441:
1424:
1421:
1395:
1394:
1367:
1366:
1365:
1359:
1343:
1341:
1340:
1335:
1333:
1332:
1299:is equal to the
1289:ring centralizer
1276:
1272:
1262:
1237:
1235:
1234:
1229:
1212:
1209:
1183:
1182:
1155:
1154:
1153:
1147:
1131:
1129:
1128:
1123:
1121:
1120:
1095:
1093:
1092:
1087:
1085:
1084:
1065:in the place of
1045:is a ring or an
1032:
1030:
1029:
1024:
1012:
1010:
1009:
1004:
993:
992:
971:
970:
915:
901:
867:
865:
864:
859:
857:
842:
840:
839:
834:
822:
820:
819:
814:
806:
791:
789:
788:
783:
772:
771:
755:
753:
752:
747:
735:
733:
732:
727:
712:
710:
709:
704:
699:
695:
688:
687:
649:
645:
600:
599:
594:
485:
483:
482:
477:
472:
468:
458:
455:
447:
446:
408:
404:
394:
391:
345:
344:
339:
238:
236:
235:
230:
208:
206:
205:
200:
189:
188:
183:
150:
148:
147:
142:
114:
112:
111:
106:
92:
91:
21:
4045:
4044:
4040:
4039:
4038:
4036:
4035:
4034:
4005:
4004:
3991:
3972:
3967:
3945:
3932:
3922:10.1090/gsm/100
3901:
3898:
3893:
3889:, Chapters 1−3.
3885:
3881:
3873:
3869:
3861:
3852:
3847:
3843:
3836:
3817:
3816:
3812:
3805:
3786:
3785:
3781:
3777:
3745:
3731:
3722:
3710:is a Lie ideal.
3697:
3668:
3662:
3637:
3623:
3617:
3608:
3598:
3590:
3581:
3556:
3534:
3500:
3490:
3475:
3455:
3440:
3436:If we define a
3398:
3392:
3381:
3375:
3353:
3343:
3322:For a subgroup
3312:
3303:
3302:if and only if
3295:
3286:
3263:
3253:
3247:
3233:
3227:
3226:if and only if
3217:
3195:
3189:
3179:
3178:
3164:
3143:is a subset of
3135:
3102:
3073:
3059:
3053:
3034:
3028:
3020:
3010:
2993:
2968:
2958:
2953:and the group N
2942:
2936:
2930:
2923:normal subgroup
2916:
2904:
2894:
2888:
2870:
2842:
2831:
2820:
2815:
2814:
2791:
2786:
2785:
2713:
2708:
2707:
2688:
2687:
2668:
2667:
2648:
2643:
2642:
2639:
2634:
2627:
2620:
2548:
2547:
2467:
2466:
2429:
2428:
2350:
2349:
2312:
2311:
2233:
2232:
2195:
2194:
2116:
2115:
2078:
2077:
2002:
2001:
1964:
1963:
1888:
1887:
1850:
1849:
1764:
1763:
1585:
1574:
1573:
1567:
1520:
1515:
1514:
1491:
1490:
1463:
1462:
1354:
1349:
1348:
1322:
1321:
1312:
1286:
1274:
1273:if and only if
1264:
1253:
1251:bracket product
1142:
1137:
1136:
1110:
1109:
1074:
1073:
1053:is a subset of
1039:
1015:
1014:
984:
962:
957:
956:
907:
893:
850:
845:
844:
825:
824:
799:
794:
793:
763:
758:
757:
738:
737:
718:
717:
676:
657:
653:
617:
613:
589:
584:
583:
520:
510:
505:set, we write C
435:
416:
412:
362:
358:
334:
329:
328:
310:
305:
215:
214:
178:
173:
172:
133:
132:
115:of elements of
83:
78:
77:
44:
37:
28:
23:
22:
15:
12:
11:
5:
4043:
4041:
4033:
4032:
4027:
4022:
4017:
4007:
4006:
4003:
4002:
3989:
3970:
3965:
3943:
3930:
3897:
3894:
3892:
3891:
3879:
3867:
3850:
3841:
3834:
3828:. p. 30.
3810:
3803:
3797:. p. 65.
3778:
3776:
3773:
3772:
3771:
3766:
3761:
3756:
3751:
3744:
3741:
3740:
3739:
3727:
3711:
3693:
3682:
3664:
3658:
3655:
3644:
3636:
3633:
3632:
3631:
3619:
3613:
3586:
3572:is said to be
3562:
3552:
3530:
3496:
3486:
3471:
3434:
3394:
3377:
3349:
3339:
3320:
3308:
3291:
3271:
3259:
3249:
3244:
3229:
3213:
3208:The center of
3206:
3191:
3166:
3160:
3137:
3131:
3112:
3098:
3087:
3069:
3055:
3049:
3046:
3030:
3016:
3006:
2992:is defined as
2964:
2954:
2938:
2926:
2921:) is always a
2912:
2900:
2890:
2885:
2869:
2866:
2848:
2845:
2841:
2837:
2834:
2830:
2826:
2823:
2797:
2794:
2773:
2770:
2767:
2764:
2761:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2719:
2716:
2695:
2675:
2654:
2651:
2638:
2635:
2633:
2630:
2625:
2618:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2544:
2543:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2426:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2309:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2192:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2075:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1961:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1840:
1839:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1757:
1756:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1592:
1588:
1584:
1581:
1566:
1563:
1542:
1539:
1536:
1530:
1524:
1500:
1472:
1451:
1450:
1439:
1436:
1433:
1430:
1427:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1393:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1364:
1358:
1331:
1308:
1282:
1239:
1238:
1227:
1224:
1221:
1218:
1215:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1181:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1152:
1146:
1119:
1083:
1038:
1035:
1022:
1002:
999:
996:
991:
987:
983:
980:
977:
974:
969:
965:
950:normal closure
856:
853:
832:
812:
809:
805:
802:
781:
778:
775:
770:
766:
745:
725:
714:
713:
702:
698:
694:
691:
686:
683:
679:
675:
672:
669:
666:
663:
660:
656:
652:
648:
644:
641:
638:
635:
632:
629:
626:
623:
620:
616:
612:
609:
606:
603:
598:
593:
579:is defined as
516:
515:) instead of C
506:
497: = {
487:
486:
475:
471:
467:
464:
461:
453:
450:
445:
442:
438:
434:
431:
428:
425:
422:
419:
415:
411:
407:
403:
400:
397:
389:
386:
383:
380:
377:
374:
371:
368:
365:
361:
357:
354:
351:
348:
343:
338:
324:is defined as
309:
306:
304:
301:
228:
225:
222:
198:
195:
192:
187:
182:
140:
104:
101:
98:
95:
90:
86:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4042:
4031:
4028:
4026:
4023:
4021:
4018:
4016:
4013:
4012:
4010:
4000:
3996:
3992:
3990:0-486-63832-4
3986:
3982:
3978:
3977:
3971:
3968:
3962:
3958:
3954:
3953:
3952:Basic Algebra
3948:
3944:
3941:
3937:
3933:
3927:
3923:
3919:
3915:
3911:
3907:
3906:
3900:
3899:
3895:
3888:
3883:
3880:
3876:
3875:Jacobson 1979
3871:
3868:
3864:
3863:Jacobson 1979
3859:
3857:
3855:
3851:
3845:
3842:
3837:
3831:
3827:
3823:
3822:
3814:
3811:
3806:
3800:
3796:
3792:
3791:
3783:
3780:
3774:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3752:
3750:
3747:
3746:
3742:
3735:
3730:
3725:
3720:
3716:
3712:
3709:
3705:
3701:
3696:
3691:
3687:
3683:
3680:
3676:
3672:
3667:
3661:
3656:
3653:
3649:
3645:
3642:
3641:
3640:
3634:
3627:
3622:
3616:
3611:
3605:
3601:
3594:
3589:
3584:
3579:
3575:
3571:
3567:
3563:
3560:
3555:
3550:
3547:pointwise is
3546:
3542:
3538:
3533:
3528:
3524:
3520:
3516:
3512:
3508:
3504:
3499:
3494:
3489:
3483:
3479:
3474:
3470:
3466:
3462:
3458:
3451:
3447:
3443:
3439:
3435:
3432:
3428:
3424:
3420:
3416:
3412:
3406:
3402:
3397:
3389:
3385:
3380:
3373:
3369:
3368:automorphisms
3365:
3361:
3357:
3352:
3347:
3342:
3337:
3333:
3329:
3325:
3321:
3316:
3311:
3306:
3299:
3294:
3289:
3284:
3280:
3276:
3272:
3267:
3262:
3257:
3252:
3245:
3241:
3237:
3232:
3225:
3224:abelian group
3221:
3216:
3212:is exactly C
3211:
3207:
3203:
3199:
3194:
3187:
3183:
3175:
3171:
3167:
3163:
3158:
3154:
3150:
3146:
3142:
3138:
3134:
3129:
3125:
3121:
3117:
3113:
3110:
3106:
3101:
3096:
3092:
3088:
3085:
3081:
3077:
3072:
3067:
3063:
3058:
3052:
3047:
3042:
3038:
3033:
3024:
3019:
3014:
3009:
3004:
3000:
2996:
2991:
2988:with a torus
2987:
2984:
2981:of a compact
2980:
2976:
2972:
2967:
2962:
2957:
2950:
2946:
2941:
2934:
2929:
2924:
2920:
2915:
2908:
2903:
2898:
2893:
2886:
2883:
2879:
2875:
2874:
2873:
2867:
2865:
2863:
2846:
2843:
2839:
2835:
2832:
2828:
2824:
2821:
2812:
2795:
2792:
2771:
2765:
2762:
2759:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2721:
2717:
2714:
2693:
2673:
2652:
2649:
2636:
2631:
2629:
2622:
2615:
2595:
2592:
2589:
2586:
2583:
2577:
2571:
2568:
2565:
2562:
2559:
2529:
2526:
2523:
2514:
2511:
2508:
2505:
2502:
2496:
2490:
2487:
2484:
2481:
2478:
2449:
2446:
2443:
2440:
2437:
2427:
2412:
2409:
2406:
2397:
2394:
2391:
2388:
2385:
2379:
2373:
2370:
2367:
2364:
2361:
2332:
2329:
2326:
2323:
2320:
2310:
2295:
2292:
2289:
2280:
2277:
2274:
2271:
2268:
2262:
2256:
2253:
2250:
2247:
2244:
2215:
2212:
2209:
2206:
2203:
2193:
2178:
2175:
2172:
2163:
2160:
2157:
2154:
2151:
2145:
2139:
2136:
2133:
2130:
2127:
2098:
2095:
2092:
2089:
2086:
2076:
2061:
2058:
2049:
2046:
2043:
2040:
2037:
2031:
2025:
2022:
2019:
2016:
2013:
1984:
1981:
1978:
1975:
1972:
1962:
1947:
1944:
1935:
1932:
1929:
1926:
1923:
1917:
1911:
1908:
1905:
1902:
1899:
1870:
1867:
1864:
1861:
1858:
1848:
1847:
1846:
1843:
1826:
1817:
1814:
1811:
1808:
1805:
1799:
1793:
1790:
1787:
1784:
1781:
1772:
1769:
1762:
1761:
1760:
1736:
1733:
1730:
1727:
1724:
1718:
1712:
1709:
1706:
1703:
1700:
1694:
1688:
1685:
1682:
1679:
1676:
1670:
1664:
1661:
1658:
1655:
1652:
1646:
1640:
1637:
1634:
1631:
1628:
1622:
1616:
1613:
1610:
1607:
1604:
1595:
1590:
1586:
1582:
1579:
1572:
1571:
1570:
1564:
1562:
1560:
1556:
1537:
1488:
1460:
1456:
1437:
1431:
1428:
1425:
1417:
1414:
1408:
1405:
1402:
1396:
1386:
1383:
1377:
1371:
1347:
1346:
1345:
1319:
1314:
1311:
1306:
1302:
1298:
1294:
1290:
1285:
1280:
1271:
1267:
1261:
1257:
1252:
1248:
1244:
1225:
1219:
1216:
1213:
1205:
1202:
1196:
1193:
1190:
1184:
1174:
1171:
1165:
1159:
1135:
1134:
1133:
1107:
1103:
1099:
1070:
1068:
1064:
1060:
1056:
1052:
1048:
1044:
1036:
1034:
1020:
997:
989:
985:
981:
975:
967:
963:
953:
951:
947:
943:
939:
935:
931:
927:
923:
919:
914:
910:
905:
900:
896:
891:
887:
883:
879:
875:
871:
854:
851:
830:
810:
807:
803:
800:
776:
768:
764:
743:
723:
700:
696:
692:
689:
684:
681:
677:
673:
670:
667:
664:
661:
658:
654:
650:
646:
642:
639:
636:
633:
630:
627:
624:
621:
618:
614:
610:
604:
596:
582:
581:
580:
578:
574:
570:
565:
563:
559:
555:
552:
548:
544:
540:
536:
532:
528:
524:
519:
514:
509:
504:
500:
496:
492:
473:
469:
465:
462:
459:
451:
448:
443:
440:
436:
432:
429:
426:
423:
420:
417:
413:
409:
405:
401:
398:
395:
387:
384:
381:
378:
375:
372:
369:
366:
363:
359:
355:
349:
341:
327:
326:
325:
323:
319:
315:
307:
302:
300:
298:
293:
291:
287:
283:
279:
275:
274:
268:
263:
261:
256:
254:
250:
246:
242:
226:
223:
220:
212:
193:
185:
170:
166:
162:
158:
154:
138:
130:
126:
122:
118:
99:
93:
88:
75:
72:
68:
65:
61:
58:(also called
57:
53:
50:, especially
49:
42:
35:
30:
19:
4030:Lie algebras
4020:Group theory
3976:Lie Algebras
3975:
3951:
3904:
3882:
3870:
3844:
3820:
3813:
3789:
3782:
3733:
3728:
3723:
3718:
3714:
3707:
3703:
3699:
3694:
3689:
3685:
3674:
3673:)) contains
3670:
3665:
3659:
3651:
3647:
3638:
3625:
3620:
3614:
3609:
3603:
3599:
3592:
3587:
3582:
3577:
3573:
3569:
3565:
3558:
3553:
3548:
3544:
3540:
3536:
3531:
3526:
3522:
3518:
3514:
3510:
3507:group action
3502:
3497:
3492:
3487:
3481:
3477:
3472:
3468:
3464:
3460:
3456:
3449:
3445:
3441:
3430:
3422:
3418:
3414:
3410:
3404:
3400:
3395:
3387:
3383:
3378:
3371:
3363:
3355:
3350:
3345:
3340:
3336:factor group
3331:
3327:
3323:
3314:
3309:
3304:
3297:
3292:
3287:
3282:
3278:
3274:
3265:
3260:
3255:
3250:
3239:
3235:
3230:
3219:
3214:
3209:
3201:
3197:
3192:
3185:
3177:
3176:is called a
3173:
3169:
3161:
3156:
3152:
3148:
3144:
3140:
3132:
3127:
3123:
3119:
3115:
3108:
3104:
3099:
3094:
3090:
3083:
3079:
3075:
3070:
3065:
3064:)) contains
3061:
3056:
3050:
3040:
3036:
3031:
3022:
3017:
3012:
3007:
3002:
2998:
2994:
2989:
2985:
2974:
2970:
2965:
2960:
2955:
2948:
2944:
2939:
2932:
2927:
2918:
2913:
2911:. In fact, C
2906:
2901:
2896:
2891:
2881:
2877:
2871:
2811:subsemigroup
2640:
2623:
2616:
2545:
1844:
1841:
1758:
1568:
1554:
1486:
1458:
1452:
1344:is given by
1317:
1315:
1309:
1304:
1300:
1296:
1292:
1288:
1283:
1278:
1269:
1265:
1259:
1255:
1246:
1242:
1240:
1105:
1071:
1066:
1062:
1058:
1054:
1050:
1042:
1040:
954:
945:
941:
937:
933:
929:
925:
921:
917:
912:
908:
903:
898:
894:
889:
885:
881:
877:
873:
869:
715:
576:
572:
568:
566:
561:
557:
553:
550:
546:
542:
538:
534:
526:
522:
517:
512:
507:
498:
494:
490:
488:
321:
317:
316:of a subset
313:
311:
294:
285:
277:
270:
264:
257:
252:
248:
240:
210:
171:of elements
164:
160:
156:
152:
124:
116:
73:
66:
59:
55:
52:group theory
45:
29:
4025:Ring theory
3887:Isaacs 2009
3568:of a group
3564:A subgroup
3332:N/C theorem
3172:of a group
3168:A subgroup
3107:) contains
3086:is abelian.
2977:. E.g. the
2862:bicommutant
1457:of the set
1098:Lie algebra
870:centralizer
547:centralizer
545:), and the
537:of a group
314:centralizer
303:Definitions
290:Lie algebra
267:ring theory
155:fixed. The
129:conjugation
76:is the set
56:centralizer
48:mathematics
4009:Categories
3896:References
3749:Commutator
3360:isomorphic
2979:Weyl group
2637:Semigroups
2632:Properties
946:S as a set
874:normalizer
569:normalizer
260:semigroups
157:normalizer
3759:Idealizer
3706:in which
3543:) fixing
3326:of group
3126:in which
2983:Lie group
2887:Clearly,
2763:∈
2737:∣
2731:∈
1557:is a Lie
1455:idealizer
1429:∈
1415:∈
1397:∣
1387:∈
1217:∈
1185:∣
1175:∈
982:⊆
916:for some
902:, but if
808:⊆
682:−
668:∣
662:∈
628:∣
622:∈
503:singleton
463:∈
441:−
427:∣
421:∈
399:∈
373:∣
367:∈
297:idealizer
245:subgroups
224:⊆
94:
60:commutant
18:Commutant
3949:(2009),
3877:, p. 57.
3865:, p. 28.
3743:See also
3692:, then N
3639:Source:
3574:C-closed
3444: :
3374:. Since
3234:(G) = Z(
3218:(G) and
3097:, then N
2947:) → Bij(
2872:Source:
2836:‴
2825:′
2809:forms a
2796:′
2718:′
2653:′
1102:Lie ring
955:Clearly
855:′
804:′
3999:0559927
3940:2472787
3721:, then
3521:in Inn(
3509:of Inn(
3495:) and C
3068:, but C
2706:; i.e.
1565:Example
1513:, then
924:, with
551:element
501:} is a
282:subring
167:is the
121:commute
62:) of a
3997:
3987:
3963:
3938:
3928:
3832:
3801:
3448:→ Inn(
3403:) = Z(
3330:, the
3222:is an
2868:Groups
1049:, and
888:is in
823:where
549:of an
535:center
531:center
269:, the
64:subset
54:, the
3775:Notes
3525:) is
3513:) on
3358:) is
3258:) = N
3005:) = N
2899:) ⊆ N
2784:Then
1559:ideal
1485:. If
1096:is a
280:is a
119:that
71:group
69:in a
3985:ISBN
3961:ISBN
3926:ISBN
3830:ISBN
3799:ISBN
3480:) =
3467:) =
3391:and
3386:) =
3277:and
3238:) =
3200:) =
3165:(S).
3136:(H).
3039:) =
2925:of N
2813:and
2641:Let
1307:in L
1100:(or
884:and
872:and
567:The
560:, Z(
541:, Z(
312:The
295:The
273:ring
243:are
3918:doi
3726:⊆ N
3713:If
3684:If
3612:= C
3585:= C
3580:if
3576:or
3561:)).
3482:xgx
3454:by
3429:of
3413:/Z(
3370:of
3348:)/C
3307:⊆ C
3290:⊆ C
3188:if
3184:of
3139:If
3114:If
3089:If
3015:)/C
2963:)/C
1461:in
1303:of
1295:in
1291:of
1275:= 0
1108:of
1072:If
1041:If
920:in
571:of
564:).
556:in
284:of
265:In
247:of
209:of
169:set
163:in
159:of
131:by
46:In
4011::
3995:MR
3993:,
3983:,
3959:,
3936:MR
3934:,
3924:,
3916:,
3908:,
3853:^
3824:.
3793:.
3663:(C
3628:))
3618:(C
3602:⊆
3551:(C
3529:(N
3463:)(
3285:,
3054:(C
2864:.
1561:.
1313:.
1270:yx
1268:=
1266:xy
1260:yx
1258:−
1256:xy
1254:=
1069:.
1033:.
952:.
913:tg
911:=
909:gs
899:sg
897:=
895:gs
521:({
292:.
262:.
255:.
3920::
3838:.
3807:.
3738:.
3736:)
3734:S
3732:(
3729:A
3724:S
3719:A
3715:S
3708:S
3704:A
3700:S
3698:(
3695:A
3690:A
3686:S
3675:S
3671:S
3669:(
3666:R
3660:R
3657:C
3654:.
3652:S
3648:S
3630:.
3626:H
3624:(
3621:G
3615:G
3610:H
3604:G
3600:S
3595:)
3593:S
3591:(
3588:G
3583:H
3570:G
3566:H
3559:S
3557:(
3554:G
3549:T
3545:S
3541:G
3537:S
3535:(
3532:G
3527:T
3523:G
3519:S
3515:G
3511:G
3503:S
3501:(
3498:G
3493:S
3491:(
3488:G
3478:g
3476:(
3473:x
3469:T
3465:g
3461:x
3459:(
3457:T
3452:)
3450:G
3446:G
3442:T
3433:.
3431:G
3423:G
3419:G
3415:G
3411:G
3407:)
3405:G
3401:G
3399:(
3396:G
3393:C
3388:G
3384:G
3382:(
3379:G
3376:N
3372:H
3364:H
3356:H
3354:(
3351:G
3346:H
3344:(
3341:G
3338:N
3328:G
3324:H
3319:.
3317:)
3315:T
3313:(
3310:G
3305:S
3300:)
3298:S
3296:(
3293:G
3288:T
3283:G
3279:T
3275:S
3270:.
3268:)
3266:a
3264:(
3261:G
3256:a
3254:(
3251:G
3248:C
3243:.
3240:G
3236:G
3231:G
3228:C
3220:G
3215:G
3210:G
3205:.
3202:H
3198:H
3196:(
3193:G
3190:N
3186:G
3174:G
3170:H
3162:G
3157:S
3153:G
3149:S
3145:G
3141:S
3133:G
3128:H
3124:G
3120:G
3116:H
3111:.
3109:H
3105:H
3103:(
3100:G
3095:G
3091:H
3084:S
3080:S
3076:S
3074:(
3071:G
3066:S
3062:S
3060:(
3057:G
3051:G
3048:C
3043:)
3041:T
3037:T
3035:(
3032:G
3029:C
3025:)
3023:T
3021:(
3018:G
3013:T
3011:(
3008:G
3003:T
3001:,
2999:G
2997:(
2995:W
2990:T
2986:G
2975:S
2971:S
2969:(
2966:G
2961:S
2959:(
2956:G
2951:)
2949:S
2945:S
2943:(
2940:G
2937:N
2933:S
2931:(
2928:G
2919:S
2917:(
2914:G
2909:)
2907:S
2905:(
2902:G
2897:S
2895:(
2892:G
2889:C
2884:.
2882:G
2878:S
2844:S
2840:=
2833:S
2829:=
2822:S
2793:S
2772:.
2769:}
2766:S
2760:s
2752:s
2749:x
2746:=
2743:x
2740:s
2734:A
2728:x
2725:{
2722:=
2715:S
2694:A
2674:S
2650:S
2626:3
2619:3
2602:}
2599:]
2596:2
2593:,
2590:3
2587:,
2584:1
2581:[
2578:,
2575:]
2572:3
2569:,
2566:2
2563:,
2560:1
2557:[
2554:{
2530:H
2527:=
2524:!
2521:}
2518:]
2515:1
2512:,
2509:3
2506:,
2503:2
2500:[
2497:,
2494:]
2491:1
2488:,
2485:2
2482:,
2479:3
2476:[
2473:{
2453:]
2450:1
2447:,
2444:2
2441:,
2438:3
2435:[
2413:H
2410:=
2407:!
2404:}
2401:]
2398:3
2395:,
2392:1
2389:,
2386:2
2383:[
2380:,
2377:]
2374:2
2371:,
2368:1
2365:,
2362:3
2359:[
2356:{
2336:]
2333:2
2330:,
2327:1
2324:,
2321:3
2318:[
2296:H
2293:=
2290:!
2287:}
2284:]
2281:1
2278:,
2275:2
2272:,
2269:3
2266:[
2263:,
2260:]
2257:1
2254:,
2251:3
2248:,
2245:2
2242:[
2239:{
2219:]
2216:1
2213:,
2210:3
2207:,
2204:2
2201:[
2179:H
2176:=
2173:!
2170:}
2167:]
2164:2
2161:,
2158:1
2155:,
2152:3
2149:[
2146:,
2143:]
2140:3
2137:,
2134:1
2131:,
2128:2
2125:[
2122:{
2102:]
2099:3
2096:,
2093:1
2090:,
2087:2
2084:[
2062:H
2059:=
2056:}
2053:]
2050:3
2047:,
2044:2
2041:,
2038:1
2035:[
2032:,
2029:]
2026:2
2023:,
2020:3
2017:,
2014:1
2011:[
2008:{
1988:]
1985:2
1982:,
1979:3
1976:,
1973:1
1970:[
1948:H
1945:=
1942:}
1939:]
1936:2
1933:,
1930:3
1927:,
1924:1
1921:[
1918:,
1915:]
1912:3
1909:,
1906:2
1903:,
1900:1
1897:[
1894:{
1874:]
1871:3
1868:,
1865:2
1862:,
1859:1
1856:[
1827:.
1824:}
1821:]
1818:2
1815:,
1812:3
1809:,
1806:1
1803:[
1800:,
1797:]
1794:3
1791:,
1788:2
1785:,
1782:1
1779:[
1776:{
1773:=
1770:H
1743:}
1740:]
1737:1
1734:,
1731:2
1728:,
1725:3
1722:[
1719:,
1716:]
1713:2
1710:,
1707:1
1704:,
1701:3
1698:[
1695:,
1692:]
1689:1
1686:,
1683:3
1680:,
1677:2
1674:[
1671:,
1668:]
1665:3
1662:,
1659:1
1656:,
1653:2
1650:[
1647:,
1644:]
1641:2
1638:,
1635:3
1632:,
1629:1
1626:[
1623:,
1620:]
1617:3
1614:,
1611:2
1608:,
1605:1
1602:[
1599:{
1596:=
1591:3
1587:S
1583:=
1580:G
1555:S
1541:)
1538:S
1535:(
1529:L
1523:N
1499:L
1487:S
1471:L
1459:S
1438:.
1435:}
1432:S
1426:s
1418:S
1412:]
1409:s
1406:,
1403:x
1400:[
1392:L
1384:x
1381:{
1378:=
1375:)
1372:S
1369:(
1363:L
1357:N
1330:L
1318:S
1310:R
1305:S
1297:R
1293:S
1284:R
1279:R
1247:R
1243:R
1226:.
1223:}
1220:S
1214:s
1206:0
1203:=
1200:]
1197:s
1194:,
1191:x
1188:[
1180:L
1172:x
1169:{
1166:=
1163:)
1160:S
1157:(
1151:L
1145:C
1118:L
1106:S
1082:L
1067:G
1063:R
1059:S
1055:R
1051:S
1043:R
1021:G
1001:)
998:S
995:(
990:G
986:N
979:)
976:S
973:(
968:G
964:C
942:S
938:S
934:S
930:s
926:t
922:S
918:t
904:g
890:S
886:s
882:S
878:g
852:G
831:S
811:G
801:G
780:)
777:S
774:(
769:G
765:N
744:G
724:S
701:,
697:}
693:S
690:=
685:1
678:g
674:S
671:g
665:G
659:g
655:{
651:=
647:}
643:g
640:S
637:=
634:S
631:g
625:G
619:g
615:{
611:=
608:)
605:S
602:(
597:G
592:N
577:G
573:S
562:g
558:G
554:g
543:G
539:G
527:a
523:a
518:G
513:a
511:(
508:G
499:a
495:S
491:G
474:,
470:}
466:S
460:s
452:s
449:=
444:1
437:g
433:s
430:g
424:G
418:g
414:{
410:=
406:}
402:S
396:s
388:g
385:s
382:=
379:s
376:g
370:G
364:g
360:{
356:=
353:)
350:S
347:(
342:G
337:C
322:G
318:S
286:R
278:R
253:S
249:G
241:S
227:G
221:S
211:G
197:)
194:S
191:(
186:G
181:N
165:G
161:S
153:S
139:g
125:S
117:G
103:)
100:S
97:(
89:G
85:C
74:G
67:S
43:.
36:.
20:)
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