267:
587:
1066:
998:
1032:
967:
712:
897:
775:
520:
351:
811:
620:
429:
380:
130:
293:
920:
854:
834:
732:
669:
644:
482:
453:
400:
317:
150:
97:
158:
1007:
The set of commuting probabilities of finite groups is reverse-well-ordered, and the reverse of its order type is known to be either
323:
714:(this result is sometimes called the 5/8 theorem) and this upper bound is sharp: there are infinitely many finite groups
1142:
Das, A. K.; Nath, R. K.; Pournaki, M. R. (2013). "A survey on the estimation of commutativity in finite groups".
534:
1037:
1317:
1164:
Hofmann, Karl H.; Russo, Francesco G. (2012). "The probability that x and y commute in a compact group".
972:
1010:
1183:
48:
1248:
1079:
929:
64:
60:
56:
674:
1293:
1275:
1199:
1173:
68:
1001:
859:
737:
1285:
1229:
1191:
1124:
490:
329:
787:
623:
596:
405:
356:
106:
16:
The probability that two uniform random elements of a finite group commute with each other
275:
1187:
905:
839:
819:
778:
717:
654:
629:
485:
467:
438:
385:
302:
135:
82:
1311:
1297:
1203:
1115:
Gustafson, W. H. (1973). "What is the
Probability that Two Group Elements Commute?".
1090:
262:{\displaystyle p(G):={\frac {1}{\#G^{2}}}\#\!\left\{(x,y)\in G^{2}\mid xy=yx\right\}}
52:
1233:
1128:
1094:
923:
100:
40:
24:
1083:
296:
44:
20:
1195:
1289:
814:
1266:
Eberhard, Sean (2015). "Commuting probabilities of finite groups".
1280:
1178:
1093:; the probability measure is then, after a renormalisation, the
1166:
Mathematical
Proceedings of the Cambridge Philosophical Society
1220:
Machale, Desmond (1976). "Commutativity in Finite Rings".
382:
is the probability that two randomly chosen elements of
1089:
The commuting probability can be defined for infinite
55:
a finite group is. It can be generalized to infinite
1040:
1013:
975:
932:
908:
862:
842:
822:
790:
740:
720:
677:
657:
632:
599:
537:
493:
470:
441:
408:
388:
359:
332:
305:
278:
161:
138:
109:
85:
1078:The commuting probability can be defined for other
1060:
1026:
992:
961:
914:
891:
848:
828:
805:
769:
726:
706:
663:
638:
614:
581:
514:
476:
447:
423:
394:
374:
345:
311:
287:
261:
144:
124:
91:
202:
132:as the averaged number of pairs of elements of
8:
1268:Bulletin of the London Mathematical Society
1279:
1177:
1050:
1045:
1039:
1018:
1012:
984:
978:
977:
974:
951:
931:
907:
881:
861:
841:
821:
789:
759:
739:
719:
696:
676:
656:
631:
598:
553:
536:
492:
469:
440:
407:
387:
358:
337:
331:
304:
277:
230:
190:
177:
160:
137:
108:
84:
51:. It can be used to measure how close to
582:{\displaystyle p(G)={\frac {k(G)}{\#G}}}
1144:Southeast Asian Bulletin of Mathematics
1107:
63:, and can also be generalized to other
1061:{\displaystyle \omega ^{\omega ^{2}}}
7:
1215:
1213:
1159:
1157:
993:{\displaystyle {\mathfrak {A}}_{5}}
979:
784:There is no uniform lower bound on
570:
279:
199:
183:
47:that two randomly chosen elements
14:
1222:The American Mathematical Monthly
1117:The American Mathematical Monthly
1027:{\displaystyle \omega ^{\omega }}
969:(this upper bound is attained by
1234:10.1080/00029890.1976.11994032
1129:10.1080/00029890.1973.11993437
942:
936:
872:
866:
813:. In fact, for every positive
800:
794:
750:
744:
687:
681:
609:
603:
565:
559:
547:
541:
503:
497:
418:
412:
369:
363:
220:
208:
171:
165:
119:
113:
1:
962:{\displaystyle p(G)\leq 1/12}
777:, the smallest one being the
1247:Baez, John C. (2018-09-16).
836:there exists a finite group
707:{\displaystyle p(G)\leq 5/8}
1334:
1196:10.1017/S0305004112000308
779:dihedral group of order 8
59:equipped with a suitable
892:{\displaystyle p(G)=1/n}
770:{\displaystyle p(G)=5/8}
33:degree of commutativity
1062:
1028:
994:
963:
916:
893:
850:
830:
807:
771:
728:
708:
665:
640:
616:
583:
516:
515:{\displaystyle p(G)=1}
478:
449:
425:
396:
376:
347:
313:
289:
263:
146:
126:
93:
23:and more precisely in
1063:
1029:
995:
964:
917:
894:
851:
831:
808:
772:
729:
709:
666:
641:
617:
584:
517:
479:
450:
433:commuting probability
426:
402:commute. That is why
397:
377:
348:
346:{\displaystyle G^{2}}
322:If one considers the
314:
290:
264:
147:
127:
94:
29:commuting probability
1080:algebraic structures
1038:
1011:
973:
930:
906:
860:
840:
820:
806:{\displaystyle p(G)}
788:
738:
718:
675:
671:is not abelian then
655:
630:
615:{\displaystyle k(G)}
597:
535:
491:
468:
439:
424:{\displaystyle p(G)}
406:
386:
375:{\displaystyle p(G)}
357:
330:
324:uniform distribution
303:
276:
159:
136:
125:{\displaystyle p(G)}
107:
83:
65:algebraic structures
37:commutativity degree
1290:10.1112/blms/bdv050
1188:2012MPCPS.153..557H
922:is not abelian but
288:{\displaystyle \#X}
61:probability measure
1058:
1024:
990:
959:
912:
889:
846:
826:
803:
767:
724:
704:
661:
636:
612:
579:
512:
474:
445:
421:
392:
372:
343:
309:
285:
259:
142:
122:
89:
1249:"The 5/8 Theorem"
1002:alternating group
915:{\displaystyle G}
849:{\displaystyle G}
829:{\displaystyle n}
727:{\displaystyle G}
664:{\displaystyle G}
639:{\displaystyle G}
624:conjugacy classes
622:is the number of
577:
477:{\displaystyle G}
464:The finite group
448:{\displaystyle G}
395:{\displaystyle G}
312:{\displaystyle X}
197:
145:{\displaystyle G}
92:{\displaystyle G}
1325:
1302:
1301:
1283:
1263:
1257:
1256:
1244:
1238:
1237:
1217:
1208:
1207:
1181:
1161:
1152:
1151:
1139:
1133:
1132:
1123:(9): 1031–1034.
1112:
1067:
1065:
1064:
1059:
1057:
1056:
1055:
1054:
1033:
1031:
1030:
1025:
1023:
1022:
999:
997:
996:
991:
989:
988:
983:
982:
968:
966:
965:
960:
955:
921:
919:
918:
913:
898:
896:
895:
890:
885:
855:
853:
852:
847:
835:
833:
832:
827:
812:
810:
809:
804:
776:
774:
773:
768:
763:
733:
731:
730:
725:
713:
711:
710:
705:
700:
670:
668:
667:
662:
645:
643:
642:
637:
621:
619:
618:
613:
588:
586:
585:
580:
578:
576:
568:
554:
521:
519:
518:
513:
483:
481:
480:
475:
454:
452:
451:
446:
430:
428:
427:
422:
401:
399:
398:
393:
381:
379:
378:
373:
352:
350:
349:
344:
342:
341:
318:
316:
315:
310:
299:of a finite set
294:
292:
291:
286:
268:
266:
265:
260:
258:
254:
235:
234:
198:
196:
195:
194:
178:
151:
149:
148:
143:
131:
129:
128:
123:
98:
96:
95:
90:
1333:
1332:
1328:
1327:
1326:
1324:
1323:
1322:
1308:
1307:
1306:
1305:
1265:
1264:
1260:
1246:
1245:
1241:
1219:
1218:
1211:
1163:
1162:
1155:
1141:
1140:
1136:
1114:
1113:
1109:
1104:
1075:
1073:Generalizations
1046:
1041:
1036:
1035:
1014:
1009:
1008:
976:
971:
970:
928:
927:
904:
903:
858:
857:
838:
837:
818:
817:
786:
785:
736:
735:
716:
715:
673:
672:
653:
652:
628:
627:
595:
594:
569:
555:
533:
532:
489:
488:
466:
465:
461:
437:
436:
404:
403:
384:
383:
355:
354:
333:
328:
327:
301:
300:
274:
273:
226:
207:
203:
186:
182:
157:
156:
152:which commute:
134:
133:
105:
104:
81:
80:
77:
17:
12:
11:
5:
1331:
1329:
1321:
1320:
1310:
1309:
1304:
1303:
1274:(5): 796–808.
1258:
1239:
1209:
1172:(3): 557–571.
1153:
1134:
1106:
1105:
1103:
1100:
1099:
1098:
1091:compact groups
1087:
1074:
1071:
1070:
1069:
1053:
1049:
1044:
1021:
1017:
1005:
987:
981:
958:
954:
950:
947:
944:
941:
938:
935:
911:
900:
888:
884:
880:
877:
874:
871:
868:
865:
845:
825:
802:
799:
796:
793:
782:
766:
762:
758:
755:
752:
749:
746:
743:
723:
703:
699:
695:
692:
689:
686:
683:
680:
660:
648:
647:
635:
611:
608:
605:
602:
591:
590:
589:
575:
572:
567:
564:
561:
558:
552:
549:
546:
543:
540:
527:
526:
523:
511:
508:
505:
502:
499:
496:
486:if and only if
473:
460:
457:
444:
431:is called the
420:
417:
414:
411:
391:
371:
368:
365:
362:
340:
336:
308:
284:
281:
270:
269:
257:
253:
250:
247:
244:
241:
238:
233:
229:
225:
222:
219:
216:
213:
210:
206:
201:
193:
189:
185:
181:
176:
173:
170:
167:
164:
141:
121:
118:
115:
112:
88:
76:
73:
15:
13:
10:
9:
6:
4:
3:
2:
1330:
1319:
1318:Finite groups
1316:
1315:
1313:
1299:
1295:
1291:
1287:
1282:
1277:
1273:
1269:
1262:
1259:
1254:
1250:
1243:
1240:
1235:
1231:
1227:
1223:
1216:
1214:
1210:
1205:
1201:
1197:
1193:
1189:
1185:
1180:
1175:
1171:
1167:
1160:
1158:
1154:
1150:(2): 161–180.
1149:
1145:
1138:
1135:
1130:
1126:
1122:
1118:
1111:
1108:
1101:
1096:
1092:
1088:
1085:
1081:
1077:
1076:
1072:
1051:
1047:
1042:
1019:
1015:
1006:
1004:of degree 5).
1003:
985:
956:
952:
948:
945:
939:
933:
925:
909:
901:
886:
882:
878:
875:
869:
863:
843:
823:
816:
797:
791:
783:
780:
764:
760:
756:
753:
747:
741:
721:
701:
697:
693:
690:
684:
678:
658:
650:
649:
633:
625:
606:
600:
592:
573:
562:
556:
550:
544:
538:
531:
530:
529:
528:
524:
509:
506:
500:
494:
487:
471:
463:
462:
458:
456:
442:
434:
415:
409:
389:
366:
360:
338:
334:
325:
320:
306:
298:
282:
255:
251:
248:
245:
242:
239:
236:
231:
227:
223:
217:
214:
211:
204:
191:
187:
179:
174:
168:
162:
155:
154:
153:
139:
116:
110:
102:
86:
74:
72:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
31:(also called
30:
26:
22:
1271:
1267:
1261:
1252:
1242:
1225:
1221:
1169:
1165:
1147:
1143:
1137:
1120:
1116:
1110:
1095:Haar measure
1084:finite rings
432:
321:
295:denotes the
271:
103:. We define
101:finite group
78:
41:finite group
36:
32:
28:
25:group theory
18:
484:is abelian
297:cardinality
45:probability
21:mathematics
1102:References
856:such that
734:such that
75:Definition
1298:119636430
1281:1411.0848
1228:: 30–32.
1204:115180549
1179:1001.4856
1048:ω
1043:ω
1020:ω
1016:ω
946:≤
691:≤
571:#
280:#
237:∣
224:∈
200:#
184:#
1312:Category
1082:such as
67:such as
1184:Bibcode
926:, then
815:integer
525:One has
459:Results
53:abelian
49:commute
43:is the
39:) of a
1296:
1253:Azimut
1202:
1000:, the
924:simple
593:where
272:where
57:groups
27:, the
1294:S2CID
1276:arXiv
1200:S2CID
1174:arXiv
99:be a
69:rings
79:Let
1286:doi
1230:doi
1192:doi
1170:153
1125:doi
1034:or
902:If
651:If
626:of
435:of
326:on
71:.
35:or
19:In
1314::
1292:.
1284:.
1272:47
1270:.
1251:.
1226:83
1224:.
1212:^
1198:.
1190:.
1182:.
1168:.
1156:^
1148:37
1146:.
1121:80
1119:.
957:12
455:.
353:,
319:.
175::=
1300:.
1288::
1278::
1255:.
1236:.
1232::
1206:.
1194::
1186::
1176::
1131:.
1127::
1097:.
1086:.
1068:.
1052:2
986:5
980:A
953:/
949:1
943:)
940:G
937:(
934:p
910:G
899:.
887:n
883:/
879:1
876:=
873:)
870:G
867:(
864:p
844:G
824:n
801:)
798:G
795:(
792:p
781:.
765:8
761:/
757:5
754:=
751:)
748:G
745:(
742:p
722:G
702:8
698:/
694:5
688:)
685:G
682:(
679:p
659:G
646:.
634:G
610:)
607:G
604:(
601:k
574:G
566:)
563:G
560:(
557:k
551:=
548:)
545:G
542:(
539:p
522:.
510:1
507:=
504:)
501:G
498:(
495:p
472:G
443:G
419:)
416:G
413:(
410:p
390:G
370:)
367:G
364:(
361:p
339:2
335:G
307:X
283:X
256:}
252:x
249:y
246:=
243:y
240:x
232:2
228:G
221:)
218:y
215:,
212:x
209:(
205:{
192:2
188:G
180:1
172:)
169:G
166:(
163:p
140:G
120:)
117:G
114:(
111:p
87:G
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