41:
1219:
1065:
1403:
864:
486:
461:
424:
1344:
1293:
1252:
788:
975:
on a (not necessarily finite) set, every element has order 2. Any such group is necessarily abelian because, since every element is its own inverse,
346:
1994:
1969:
1940:
1912:
296:
781:
291:
2019:
1214:{\displaystyle (\mathbb {Z} /p\mathbb {Z} )^{n}\cong \langle e_{1},\ldots ,e_{n}\mid e_{i}^{p}=1,\ e_{i}e_{j}=e_{j}e_{i}\rangle }
707:
994:. Such a group (also called a Boolean group), generalizes the Klein four-group example to an arbitrary number of components.
2041:
1782:
It can also be of interest to go beyond prime order components to prime-power order. Consider an elementary abelian group
774:
939:. (Note that in the finite case the direct product and direct sum coincide, but this is not so in the infinite case.)
391:
205:
1722:(as is true for any vector space). This in fact characterizes elementary abelian groups among all finite groups: if
1715:
1381:
921:
589:
200:
88:
1375:. Note that an elementary abelian group does not in general have a distinguished basis: choice of isomorphism
1053:
2046:
739:
529:
613:
1672:
1441:
972:
868:
818:
553:
541:
159:
93:
1432:, in particular that it has scalar multiplication in addition to (vector/group) addition. However,
1855:
128:
23:
469:
444:
407:
1900:
910:
113:
85:
2051:
2015:
1990:
1965:
1936:
1908:
684:
518:
361:
255:
960:. Addition is performed componentwise, taking the result modulo 2. For instance,
1875:
1863:
1329:
1278:
1237:
965:
669:
661:
653:
645:
637:
625:
565:
505:
495:
337:
279:
154:
123:
753:
746:
732:
689:
577:
500:
330:
244:
184:
64:
863:
elements, and conversely every such vector space is an elementary abelian group. By the
1572:
760:
696:
386:
366:
303:
268:
189:
179:
164:
149:
103:
80:
2035:
1880:
814:
679:
601:
435:
308:
174:
1767:
1629:
1614:
1363:
1304:
902:
852:
822:
806:
534:
233:
222:
169:
144:
139:
98:
69:
32:
1770:, which is necessarily invariant under all automorphisms, and thus equals all of
1957:
856:
802:
932:
701:
429:
522:
40:
59:
1858:
are extensions of elementary abelian groups by a cyclic group of order
830:
401:
315:
1016:
is the least possible number of generators. In particular, the set
840:= 2 (that is, an elementary abelian 2-group) is sometimes called a
16:
Commutative group in which all nonzero elements have the same order
825:, and the elementary abelian groups in which the common order is
871:, every finite elementary abelian group must be of the form (
817:
in which all elements other than the identity have the same
1052:
Every finite elementary abelian group has a fairly simple
1049:
th component and 0 elsewhere, is a minimal generating set.
1935:. Springer Science & Business Media. p. 88.
1384:
1332:
1281:
1240:
1068:
927:
In general, a (possibly infinite) elementary abelian
886:
a non-negative integer (sometimes called the group's
472:
447:
410:
1964:. Springer Science & Business Media. p. 6.
1498:(considered as an integer with 0 ≤
865:
classification of finitely generated abelian groups
1397:
1338:
1287:
1246:
1213:
480:
455:
418:
1621:can be considered as a linear transformation of
1605:can be considered as a group homomorphism from
1597:extends uniquely to a linear transformation of
1265:) is a finite elementary abelian group. Since
867:, or by the fact that every vector space has a
1754:have the same (necessarily prime) order. Then
782:
8:
1547:} as described in the examples, if we take {
1419:To the observant reader, it may appear that
1208:
1103:
1448:corresponds to repeated addition, and this
1926:
1924:
916:), and the superscript notation means the
789:
775:
227:
53:
18:
2014:. New York: Harper & Row. p. 8.
1452:-module structure is consistent with the
1398:{\displaystyle {\overset {\cong }{\to }}}
1385:
1383:
1331:
1280:
1239:
1202:
1192:
1179:
1169:
1147:
1142:
1129:
1110:
1094:
1086:
1085:
1077:
1073:
1072:
1067:
474:
473:
471:
449:
448:
446:
412:
411:
409:
1892:
345:
111:
21:
1742:is elementary abelian. (Proof: if Aut(
347:Classification of finite simple groups
1527:As a finite-dimensional vector space
7:
1907:. Courier Corporation. p. 142.
1416:) corresponds to a choice of basis.
1750:, then all nonidentity elements of
1617:) and likewise any endomorphism of
2010:Gorenstein, Daniel (1968). "1.2".
1830:isomorphic cyclic groups of order
1428:has more structure than the group
14:
1987:Infinite Abelian Groups. Volume I
1778:A generalisation to higher orders
1461:scalar multiplication. That is,
1436:as an abelian group has a unique
1962:Introduction to Boolean Algebras
1726:is a finite group with identity
1628:If we restrict our attention to
39:
1814:) is an abelian group of type (
821:. This common order must be a
1989:. Academic Press. p. 43.
1444:structure where the action of
1387:
1091:
1069:
971:In the group generated by the
948:The elementary abelian group (
909:(or equivalently the integers
708:Infinite dimensional Lie group
1:
1826:) i.e. the direct product of
1477: + ... +
958:{(0,0), (0,1), (1,0), (1,1)}
481:{\displaystyle \mathbb {Z} }
456:{\displaystyle \mathbb {Z} }
419:{\displaystyle \mathbb {Z} }
1834:, of which groups of type (
206:List of group theory topics
2068:
1695:The automorphism group GL(
935:of cyclic groups of order
1933:A Course on Finite Groups
1862:and are analogous to the
1575:we have that the mapping
847:Every elementary abelian
829:are a particular kind of
1762:-group. It follows that
1358:can be considered as an
922:direct product of groups
811:elementary abelian group
324:Elementary abelian group
201:Glossary of group theory
1746:) acts transitively on
1734:) acts transitively on
1683:invertible matrices on
943:Examples and properties
1846:) are a special case.
1399:
1340:
1339:{\displaystyle \cong }
1289:
1288:{\displaystyle \cong }
1248:
1247:{\displaystyle \cong }
1227:Vector space structure
1215:
964:. This is in fact the
740:Linear algebraic group
482:
457:
420:
1400:
1341:
1290:
1249:
1216:
962:(1,0) + (1,1) = (0,1)
956:) has four elements:
483:
458:
421:
2042:Abelian group theory
1905:The Theory of Groups
1856:extra special groups
1673:general linear group
1382:
1330:
1279:
1238:
1066:
973:symmetric difference
836:. A group for which
470:
445:
408:
1625:as a vector space.
1519:-module structure.
1152:
1054:finite presentation
114:Group homomorphisms
24:Algebraic structure
1931:H.E. Rose (2009).
1901:Hans J. Zassenhaus
1679: ×
1523:Automorphism group
1395:
1336:
1311:elements, we have
1285:
1244:
1211:
1138:
1008:) is generated by
805:, specifically in
590:Special orthogonal
478:
453:
416:
297:Lagrange's theorem
1996:978-0-08-087348-0
1985:L. Fuchs (1970).
1971:978-0-387-40293-2
1942:978-1-84882-889-6
1914:978-0-486-16568-4
1802:) for some prime
1766:has a nontrivial
1393:
1164:
799:
798:
374:
373:
256:Alternating group
213:
212:
2059:
2026:
2025:
2007:
2001:
2000:
1982:
1976:
1975:
1953:
1947:
1946:
1928:
1919:
1918:
1897:
1876:Elementary group
1864:Heisenberg group
1808:homocyclic group
1502: <
1404:
1402:
1401:
1396:
1394:
1386:
1345:
1343:
1342:
1337:
1294:
1292:
1291:
1286:
1253:
1251:
1250:
1245:
1220:
1218:
1217:
1212:
1207:
1206:
1197:
1196:
1184:
1183:
1174:
1173:
1162:
1151:
1146:
1134:
1133:
1115:
1114:
1099:
1098:
1089:
1081:
1076:
1035:
966:Klein four-group
963:
959:
791:
784:
777:
733:Algebraic groups
506:Hyperbolic group
496:Arithmetic group
487:
485:
484:
479:
477:
462:
460:
459:
454:
452:
425:
423:
422:
417:
415:
338:Schur multiplier
292:Cauchy's theorem
280:Quaternion group
228:
54:
43:
30:
19:
2067:
2066:
2062:
2061:
2060:
2058:
2057:
2056:
2032:
2031:
2030:
2029:
2022:
2009:
2008:
2004:
1997:
1984:
1983:
1979:
1972:
1956:Steven Givant;
1955:
1954:
1950:
1943:
1930:
1929:
1922:
1915:
1899:
1898:
1894:
1889:
1872:
1852:
1780:
1713:
1704:
1691:
1670:
1661:
1596:
1587:
1562:
1553:
1546:
1537:
1525:
1518:
1497:
1460:
1427:
1380:
1379:
1374:
1366:over the field
1353:
1328:
1327:
1302:
1277:
1276:
1236:
1235:
1229:
1198:
1188:
1175:
1165:
1125:
1106:
1090:
1064:
1063:
1045:has a 1 in the
1044:
1033:
1024:
1017:
961:
957:
945:
795:
766:
765:
754:Abelian variety
747:Reductive group
735:
725:
724:
723:
722:
673:
665:
657:
649:
641:
614:Special unitary
525:
511:
510:
492:
491:
468:
467:
443:
442:
406:
405:
397:
396:
387:Discrete groups
376:
375:
331:Frobenius group
276:
263:
252:
245:Symmetric group
241:
225:
215:
214:
65:Normal subgroup
51:
31:
22:
17:
12:
11:
5:
2065:
2063:
2055:
2054:
2049:
2044:
2034:
2033:
2028:
2027:
2020:
2002:
1995:
1977:
1970:
1948:
1941:
1920:
1913:
1891:
1890:
1888:
1885:
1884:
1883:
1878:
1871:
1868:
1851:
1850:Related groups
1848:
1779:
1776:
1730:such that Aut(
1709:
1700:
1687:
1666:
1657:
1592:
1583:
1573:linear algebra
1558:
1551:
1542:
1535:
1524:
1521:
1514:
1493:
1456:
1423:
1392:
1389:
1370:
1349:
1335:
1298:
1284:
1243:
1228:
1225:
1224:
1223:
1222:
1221:
1210:
1205:
1201:
1195:
1191:
1187:
1182:
1178:
1172:
1168:
1161:
1158:
1155:
1150:
1145:
1141:
1137:
1132:
1128:
1124:
1121:
1118:
1113:
1109:
1105:
1102:
1097:
1093:
1088:
1084:
1080:
1075:
1071:
1058:
1057:
1050:
1040:
1029:
1022:
1012:elements, and
995:
969:
944:
941:
797:
796:
794:
793:
786:
779:
771:
768:
767:
764:
763:
761:Elliptic curve
757:
756:
750:
749:
743:
742:
736:
731:
730:
727:
726:
721:
720:
717:
714:
710:
706:
705:
704:
699:
697:Diffeomorphism
693:
692:
687:
682:
676:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
634:
633:
622:
621:
610:
609:
598:
597:
586:
585:
574:
573:
562:
561:
554:Special linear
550:
549:
542:General linear
538:
537:
532:
526:
517:
516:
513:
512:
509:
508:
503:
498:
490:
489:
476:
464:
451:
438:
436:Modular groups
434:
433:
432:
427:
414:
398:
395:
394:
389:
383:
382:
381:
378:
377:
372:
371:
370:
369:
364:
359:
356:
350:
349:
343:
342:
341:
340:
334:
333:
327:
326:
321:
312:
311:
309:Hall's theorem
306:
304:Sylow theorems
300:
299:
294:
286:
285:
284:
283:
277:
272:
269:Dihedral group
265:
264:
259:
253:
248:
242:
237:
226:
221:
220:
217:
216:
211:
210:
209:
208:
203:
195:
194:
193:
192:
187:
182:
177:
172:
167:
162:
160:multiplicative
157:
152:
147:
142:
134:
133:
132:
131:
126:
118:
117:
109:
108:
107:
106:
104:Wreath product
101:
96:
91:
89:direct product
83:
81:Quotient group
75:
74:
73:
72:
67:
62:
52:
49:
48:
45:
44:
36:
35:
15:
13:
10:
9:
6:
4:
3:
2:
2064:
2053:
2050:
2048:
2047:Finite groups
2045:
2043:
2040:
2039:
2037:
2023:
2021:0-8218-4342-7
2017:
2013:
2012:Finite Groups
2006:
2003:
1998:
1992:
1988:
1981:
1978:
1973:
1967:
1963:
1959:
1952:
1949:
1944:
1938:
1934:
1927:
1925:
1921:
1916:
1910:
1906:
1902:
1896:
1893:
1886:
1882:
1881:Hamming space
1879:
1877:
1874:
1873:
1869:
1867:
1865:
1861:
1857:
1849:
1847:
1845:
1841:
1837:
1833:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1801:
1797:
1793:
1789:
1785:
1777:
1775:
1773:
1769:
1765:
1761:
1757:
1753:
1749:
1745:
1741:
1737:
1733:
1729:
1725:
1721:
1717:
1712:
1708:
1703:
1698:
1693:
1690:
1686:
1682:
1678:
1674:
1669:
1665:
1660:
1655:
1651:
1647:
1643:
1639:
1635:
1631:
1630:automorphisms
1626:
1624:
1620:
1616:
1612:
1608:
1604:
1601:. Each such
1600:
1595:
1591:
1586:
1582:
1578:
1574:
1570:
1566:
1561:
1557:
1550:
1545:
1541:
1534:
1531:has a basis {
1530:
1522:
1520:
1517:
1513:
1509:
1505:
1501:
1496:
1492:
1488:
1485:times) where
1484:
1480:
1476:
1473: +
1472:
1468:
1464:
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1431:
1426:
1422:
1417:
1415:
1412:
1408:
1390:
1378:
1373:
1369:
1365:
1362:-dimensional
1361:
1357:
1352:
1348:
1333:
1325:
1322:
1318:
1314:
1310:
1306:
1301:
1297:
1282:
1275:
1272:
1268:
1264:
1261:
1257:
1241:
1234:
1226:
1203:
1199:
1193:
1189:
1185:
1180:
1176:
1170:
1166:
1159:
1156:
1153:
1148:
1143:
1139:
1135:
1130:
1126:
1122:
1119:
1116:
1111:
1107:
1100:
1095:
1082:
1078:
1062:
1061:
1060:
1059:
1055:
1051:
1048:
1043:
1039:
1032:
1028:
1021:
1015:
1011:
1007:
1004:
1000:
996:
993:
989:
986:
982:
978:
974:
970:
967:
955:
951:
947:
946:
942:
940:
938:
934:
930:
925:
923:
919:
915:
912:
908:
904:
900:
897:
893:
889:
885:
881:
878:
874:
870:
866:
862:
858:
854:
850:
845:
843:
842:Boolean group
839:
835:
833:
828:
824:
820:
816:
815:abelian group
812:
808:
804:
792:
787:
785:
780:
778:
773:
772:
770:
769:
762:
759:
758:
755:
752:
751:
748:
745:
744:
741:
738:
737:
734:
729:
728:
718:
715:
712:
711:
709:
703:
700:
698:
695:
694:
691:
688:
686:
683:
681:
678:
677:
674:
668:
666:
660:
658:
652:
650:
644:
642:
636:
635:
631:
627:
624:
623:
619:
615:
612:
611:
607:
603:
600:
599:
595:
591:
588:
587:
583:
579:
576:
575:
571:
567:
564:
563:
559:
555:
552:
551:
547:
543:
540:
539:
536:
533:
531:
528:
527:
524:
520:
515:
514:
507:
504:
502:
499:
497:
494:
493:
465:
440:
439:
437:
431:
428:
403:
400:
399:
393:
390:
388:
385:
384:
380:
379:
368:
365:
363:
360:
357:
354:
353:
352:
351:
348:
344:
339:
336:
335:
332:
329:
328:
325:
322:
320:
318:
314:
313:
310:
307:
305:
302:
301:
298:
295:
293:
290:
289:
288:
287:
281:
278:
275:
270:
267:
266:
262:
257:
254:
251:
246:
243:
240:
235:
232:
231:
230:
229:
224:
223:Finite groups
219:
218:
207:
204:
202:
199:
198:
197:
196:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
137:
136:
135:
130:
127:
125:
122:
121:
120:
119:
116:
115:
110:
105:
102:
100:
97:
95:
92:
90:
87:
84:
82:
79:
78:
77:
76:
71:
68:
66:
63:
61:
58:
57:
56:
55:
50:Basic notions
47:
46:
42:
38:
37:
34:
29:
25:
20:
2011:
2005:
1986:
1980:
1961:
1951:
1932:
1904:
1895:
1859:
1853:
1843:
1839:
1835:
1831:
1827:
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1795:
1791:
1787:
1783:
1781:
1771:
1763:
1759:
1755:
1751:
1747:
1743:
1739:
1735:
1731:
1727:
1723:
1719:
1716:transitively
1710:
1706:
1701:
1696:
1694:
1688:
1684:
1680:
1676:
1667:
1663:
1658:
1653:
1649:
1645:
1641:
1637:
1636:we have Aut(
1633:
1627:
1622:
1618:
1615:endomorphism
1610:
1606:
1602:
1598:
1593:
1589:
1584:
1580:
1576:
1568:
1567:elements of
1564:
1563:} to be any
1559:
1555:
1548:
1543:
1539:
1532:
1528:
1526:
1515:
1511:
1507:
1503:
1499:
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1437:
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1424:
1420:
1418:
1413:
1410:
1406:
1376:
1371:
1367:
1364:vector space
1359:
1355:
1350:
1346:
1323:
1320:
1316:
1312:
1308:
1305:finite field
1299:
1295:
1273:
1270:
1266:
1262:
1259:
1255:
1232:
1230:
1046:
1041:
1037:
1030:
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1013:
1009:
1005:
1002:
998:
991:
987:
984:
980:
976:
953:
949:
936:
931:-group is a
928:
926:
917:
913:
906:
903:cyclic group
901:denotes the
898:
895:
891:
887:
883:
879:
876:
872:
860:
853:vector space
851:-group is a
848:
846:
841:
837:
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826:
823:prime number
810:
807:group theory
800:
629:
617:
605:
593:
581:
569:
557:
545:
323:
316:
273:
260:
249:
238:
234:Cyclic group
112:
99:Free product
70:Group action
33:Group theory
28:Group theory
27:
1958:Paul Halmos
857:prime field
803:mathematics
519:Topological
358:alternating
2036:Categories
1887:References
1656:= 0 } = GL
1571:, then by
1510:a natural
933:direct sum
626:Symplectic
566:Orthogonal
523:Lie groups
430:Free group
155:continuous
94:Direct sum
1903:(1999) .
1810:(of rank
1786:to be of
1391:≅
1388:→
1334:≅
1283:≅
1242:≅
1209:⟩
1136:∣
1120:…
1104:⟨
1101:≅
905:of order
890:). Here,
855:over the
690:Conformal
578:Euclidean
185:nilpotent
2052:P-groups
1960:(2009).
1870:See also
1648:→
1644: :
1506:) gives
1354:, hence
1231:Suppose
1036:, where
685:Poincaré
530:Solenoid
402:Integers
392:Lattices
367:sporadic
362:Lie type
190:solvable
180:dihedral
165:additive
150:infinite
60:Subgroup
1748:G \ {e}
1738:, then
1736:G \ {e}
1720:V \ {0}
1714:) acts
1671:), the
1554:, ...,
1538:, ...,
1465:⋅
1025:, ...,
680:Lorentz
602:Unitary
501:Lattice
441:PSL(2,
175:abelian
86:(Semi-)
2018:
1993:
1968:
1939:
1911:
1768:center
1699:) = GL
1652:| ker
1640:) = {
1442:module
1303:, the
1163:
920:-fold
882:) for
834:-group
813:is an
535:Circle
466:SL(2,
355:cyclic
319:-group
170:cyclic
145:finite
140:simple
124:kernel
1842:,...,
1822:,...,
1798:,...,
1758:is a
869:basis
859:with
819:order
809:, an
719:Sp(∞)
716:SU(∞)
129:image
2016:ISBN
1991:ISBN
1966:ISBN
1937:ISBN
1909:ISBN
1854:The
1806:. A
1788:type
1613:(an
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983:) =
888:rank
713:O(∞)
702:Loop
521:and
1774:.)
1718:on
1675:of
1632:of
1609:to
1489:in
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1307:of
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911:mod
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404:(
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236:Z
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