25:
533:
501:
246:
211:
635:
420:
583:
169:
553:
352:
321:
294:
472:
440:
394:
374:
189:
35:
846:
702:
555:
If the group operation is called multiplication then 1 can be a notation for the trivial group. Combining these leads to the
50:
93:
65:
72:
695:
757:
79:
1157:
1188:
1042:
607:
61:
1028:
688:
645:
258:
is also a group since its only element is its own inverse, and is hence the same as the trivial group.
980:
506:
477:
1004:
998:
762:
651:
737:
711:
127:
937:
753:
747:
592:
216:
194:
1099:
860:
616:
1167:
1104:
1079:
1071:
1063:
1055:
1047:
1034:
1016:
1010:
611:
139:
399:
86:
1120:
954:
823:
732:
600:
562:
145:
538:
334:
303:
276:
1125:
967:
920:
913:
906:
899:
871:
838:
808:
803:
793:
742:
596:
457:
425:
379:
359:
174:
1182:
1135:
1094:
1022:
943:
928:
876:
783:
265:, which has no elements, hence lacks an identity element, and so cannot be a group.
1162:
1089:
889:
851:
798:
788:
778:
719:
556:
535:
If the group operation is called addition, the trivial group is usually denoted by
451:
666:
818:
813:
588:
115:
24:
1130:
131:
990:
671:
262:
727:
297:
559:
in which the addition and multiplication operations are identical and
376:
has no nontrivial proper subgroups" refers to the only subgroups of
42:
684:
138:
trivial group. The single element of the trivial group is the
18:
191:
depending on the context. If the group operation is denoted
680:
296:
the group consisting of only the identity element is a
46:
619:
565:
541:
509:
480:
460:
428:
402:
382:
362:
337:
306:
279:
219:
197:
177:
148:
130:
consisting of a single element. All such groups are
1113:
989:
837:
771:
718:
629:
577:
547:
527:
495:
466:
434:
414:
388:
368:
346:
315:
288:
240:
205:
183:
163:
648: – Algebraic structure with only one element
696:
8:
409:
403:
323:and, being the trivial group, is called the
51:introducing citations to additional sources
703:
689:
681:
620:
618:
564:
540:
516:
511:
508:
487:
482:
479:
459:
427:
401:
381:
361:
336:
305:
278:
218:
202:
198:
196:
176:
147:
41:Relevant discussion may be found on the
665:Rowland, Todd & Weisstein, Eric W.
261:The trivial group is distinct from the
847:Classification of finite simple groups
142:and so it is usually denoted as such:
606:The trivial group can be made a (bi-)
7:
16:Group consisting of only one element
512:
483:
14:
610:by equipping it with the trivial
528:{\displaystyle \mathrm {C} _{1}.}
587:The trivial group serves as the
496:{\displaystyle \mathrm {Z} _{1}}
34:relies largely or entirely on a
23:
1:
474:; as such it may be denoted
356:The term, when referred to "
1114:Infinite dimensional groups
241:{\displaystyle e\cdot e=e.}
1205:
206:{\displaystyle \,\cdot \,}
1153:
134:, so one often speaks of
1017:Special orthogonal group
630:{\displaystyle \,\leq .}
595:, meaning it is both an
396:being the trivial group
1043:Exceptional Lie groups
631:
579:
549:
529:
497:
468:
436:
416:
390:
370:
348:
317:
290:
250:The similarly defined
242:
213:then it is defined by
207:
185:
165:
1029:Special unitary group
646:Zero object (algebra)
632:
580:
550:
530:
498:
469:
450:The trivial group is
437:
417:
415:{\displaystyle \{e\}}
391:
371:
349:
318:
291:
243:
208:
186:
166:
1126:Diffeomorphism group
1005:Special linear group
999:General linear group
652:List of small groups
617:
578:{\displaystyle 0=1.}
563:
539:
507:
478:
458:
426:
400:
380:
360:
335:
304:
277:
217:
195:
175:
164:{\displaystyle 0,1,}
146:
47:improve this article
951:Other finite groups
738:Commutator subgroup
981:Rubik's Cube group
938:Baby monster group
748:Group homomorphism
627:
593:category of groups
575:
548:{\displaystyle 0.}
545:
525:
493:
464:
432:
412:
386:
366:
347:{\displaystyle G.}
344:
316:{\displaystyle G,}
313:
289:{\displaystyle G,}
286:
238:
203:
181:
161:
1176:
1175:
861:Alternating group
467:{\displaystyle 1}
435:{\displaystyle G}
389:{\displaystyle G}
369:{\displaystyle G}
184:{\displaystyle e}
112:
111:
97:
1196:
1168:Abstract algebra
1105:Quaternion group
1035:Symplectic group
1011:Orthogonal group
705:
698:
691:
682:
677:
676:
636:
634:
633:
628:
612:non-strict order
584:
582:
581:
576:
554:
552:
551:
546:
534:
532:
531:
526:
521:
520:
515:
502:
500:
499:
494:
492:
491:
486:
473:
471:
470:
465:
441:
439:
438:
433:
421:
419:
418:
413:
395:
393:
392:
387:
375:
373:
372:
367:
353:
351:
350:
345:
329:
328:
327:trivial subgroup
322:
320:
319:
314:
295:
293:
292:
287:
273:Given any group
256:
255:
247:
245:
244:
239:
212:
210:
209:
204:
190:
188:
187:
182:
170:
168:
167:
162:
140:identity element
107:
104:
98:
96:
55:
27:
19:
1204:
1203:
1199:
1198:
1197:
1195:
1194:
1193:
1179:
1178:
1177:
1172:
1149:
1121:Conformal group
1109:
1083:
1075:
1067:
1059:
1051:
985:
977:
964:
955:Symmetric group
934:
924:
917:
910:
903:
895:
886:
882:
872:Sporadic groups
866:
857:
839:Discrete groups
833:
824:Wallpaper group
804:Solvable groups
772:Types of groups
767:
733:Normal subgroup
714:
709:
667:"Trivial Group"
664:
663:
660:
642:
615:
614:
601:terminal object
561:
560:
537:
536:
510:
505:
504:
481:
476:
475:
456:
455:
448:
424:
423:
398:
397:
378:
377:
358:
357:
333:
332:
326:
325:
302:
301:
275:
274:
271:
253:
252:
215:
214:
193:
192:
173:
172:
144:
143:
108:
102:
99:
62:"Trivial group"
56:
54:
40:
28:
17:
12:
11:
5:
1202:
1200:
1192:
1191:
1181:
1180:
1174:
1173:
1171:
1170:
1165:
1160:
1154:
1151:
1150:
1148:
1147:
1144:
1141:
1138:
1133:
1128:
1123:
1117:
1115:
1111:
1110:
1108:
1107:
1102:
1100:Poincaré group
1097:
1092:
1086:
1085:
1081:
1077:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1039:
1038:
1032:
1026:
1020:
1014:
1008:
1002:
995:
993:
987:
986:
984:
983:
978:
973:
968:Dihedral group
965:
960:
952:
948:
947:
941:
935:
932:
926:
922:
915:
908:
901:
896:
893:
887:
884:
880:
874:
868:
867:
864:
858:
855:
849:
843:
841:
835:
834:
832:
831:
826:
821:
816:
811:
809:Symmetry group
806:
801:
796:
794:Infinite group
791:
786:
784:Abelian groups
781:
775:
773:
769:
768:
766:
765:
760:
758:direct product
750:
745:
743:Quotient group
740:
735:
730:
724:
722:
716:
715:
710:
708:
707:
700:
693:
685:
679:
678:
659:
656:
655:
654:
649:
641:
638:
626:
623:
597:initial object
574:
571:
568:
544:
524:
519:
514:
490:
485:
463:
447:
444:
431:
422:and the group
411:
408:
405:
385:
365:
343:
340:
312:
309:
285:
282:
270:
267:
254:trivial monoid
237:
234:
231:
228:
225:
222:
201:
180:
160:
157:
154:
151:
137:
110:
109:
45:. Please help
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
1201:
1190:
1189:Finite groups
1187:
1186:
1184:
1169:
1166:
1164:
1161:
1159:
1156:
1155:
1152:
1145:
1142:
1139:
1137:
1136:Quantum group
1134:
1132:
1129:
1127:
1124:
1122:
1119:
1118:
1116:
1112:
1106:
1103:
1101:
1098:
1096:
1095:Lorentz group
1093:
1091:
1088:
1087:
1084:
1078:
1076:
1070:
1068:
1062:
1060:
1054:
1052:
1046:
1044:
1041:
1040:
1036:
1033:
1030:
1027:
1024:
1023:Unitary group
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
996:
994:
992:
988:
982:
979:
976:
972:
969:
966:
963:
959:
956:
953:
950:
949:
945:
944:Monster group
942:
939:
936:
930:
929:Fischer group
927:
925:
918:
911:
904:
898:Janko groups
897:
891:
888:
878:
877:Mathieu group
875:
873:
870:
869:
862:
859:
853:
850:
848:
845:
844:
842:
840:
836:
830:
829:Trivial group
827:
825:
822:
820:
817:
815:
812:
810:
807:
805:
802:
800:
799:Simple groups
797:
795:
792:
790:
789:Cyclic groups
787:
785:
782:
780:
779:Finite groups
777:
776:
774:
770:
764:
761:
759:
755:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
725:
723:
721:
720:Basic notions
717:
713:
706:
701:
699:
694:
692:
687:
686:
683:
674:
673:
668:
662:
661:
657:
653:
650:
647:
644:
643:
639:
637:
624:
621:
613:
609:
608:ordered group
604:
602:
598:
594:
590:
585:
572:
569:
566:
558:
542:
522:
517:
488:
461:
453:
445:
443:
429:
406:
383:
363:
354:
341:
338:
330:
310:
307:
299:
283:
280:
268:
266:
264:
259:
257:
248:
235:
232:
229:
226:
223:
220:
199:
178:
158:
155:
152:
149:
141:
135:
133:
129:
125:
121:
120:trivial group
117:
106:
95:
92:
88:
85:
81:
78:
74:
71:
67:
64: –
63:
59:
58:Find sources:
52:
48:
44:
38:
37:
36:single source
32:This article
30:
26:
21:
20:
1163:Applications
1090:Circle group
974:
970:
961:
957:
890:Conway group
852:Cyclic group
828:
670:
605:
586:
557:trivial ring
449:
355:
324:
272:
260:
251:
249:
123:
119:
113:
100:
90:
83:
76:
69:
57:
33:
819:Point group
814:Space group
589:zero object
269:Definitions
116:mathematics
1131:Loop group
991:Lie groups
763:direct sum
658:References
446:Properties
132:isomorphic
124:zero group
73:newspapers
672:MathWorld
622:≤
454:of order
263:empty set
224:⋅
200:⋅
43:talk page
1183:Category
728:Subgroup
640:See also
442:itself.
298:subgroup
103:May 2024
1158:History
591:in the
87:scholar
933:22..24
885:22..24
881:11..12
712:Groups
599:and a
452:cyclic
89:
82:
75:
68:
60:
1146:Sp(∞)
1143:SU(∞)
1037:Sp(n)
1031:SU(n)
1019:SO(n)
1007:SL(n)
1001:GL(n)
754:Semi-
128:group
126:is a
94:JSTOR
80:books
1140:O(∞)
1025:U(n)
1013:O(n)
894:1..3
118:, a
66:news
503:or
331:of
300:of
171:or
136:the
122:or
114:In
49:by
1185::
919:,
912:,
905:,
892:Co
883:,M
756:)
669:.
603:.
573:1.
543:0.
1082:8
1080:E
1074:7
1072:E
1066:6
1064:E
1058:4
1056:F
1050:2
1048:G
975:n
971:D
962:n
958:S
946:M
940:B
931:F
923:4
921:J
916:3
914:J
909:2
907:J
902:1
900:J
879:M
865:n
863:A
856:n
854:Z
752:(
704:e
697:t
690:v
675:.
625:.
570:=
567:0
523:.
518:1
513:C
489:1
484:Z
462:1
430:G
410:}
407:e
404:{
384:G
364:G
342:.
339:G
311:,
308:G
284:,
281:G
236:.
233:e
230:=
227:e
221:e
179:e
159:,
156:1
153:,
150:0
105:)
101:(
91:·
84:·
77:·
70:·
53:.
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
