323:
showed that any non-regular transitive permutation group with a regular subgroup is a Zappa–Szép product of the regular subgroup and a point stabilizer. He gives PSL(2,11) and the alternating group of degree 5 as examples, and of course every alternating group of prime degree is an example. This
1420:
1491:
324:
same paper gives a number of examples of groups which cannot be realized as Zappa–Szép products of proper subgroups, such as the quaternion group and the alternating group of degree 6.
1572:
1846:
1511:
1273:
1691:
1660:
316:-subgroup, and in fact that the group is a (multiple factor) Zappa–Szép product of a certain set of representatives of its Sylow subgroups.
86:(1950) although it was independently studied by others including B.H. Neumann (1935), G.A. Miller (1935), and J.A. de Séguier (1904).
332:
As with the direct and semidirect products, there is an external version of the Zappa–Szép product for groups which are not known
1425:
901:
877:
1629:
1516:
71:
249:
1946:
1796:
Szép, J. (1950), "On the structure of groups which can be represented as the product of two subgroups",
1831:
Zappa, G. (1940), "Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili traloro",
1870:
1757:
1736:
222:
212:
320:
63:
1925:
1907:
1886:
1860:
1726:
75:
1840:
1785:
1705:
1687:
1656:
1175:
1020:
417:
216:
1917:
1878:
1819:
1775:
1765:
238:
99:
1701:
1748:
Miller, G. A. (1935), "Groups which are the products of two permutable proper subgroups",
1697:
1683:
1675:
1239:
256:
83:
1874:
1761:
1740:
1780:
1648:
1496:
242:
226:
1810:
Takeuchi, M. (1981), "Matched pairs of groups and bismash products of Hopf algebras",
1940:
305:
301:
279:
1929:
1890:
35:
17:
1415:{\displaystyle (h_{1}k_{1})(h_{2}k_{2})=(h_{1}k_{1}h_{2}k_{1}^{-1})(k_{1}k_{2})}
995:
denote each group's identity element) and suppose there exist mappings α :
297:
259:
79:
31:
1882:
1823:
864:
More concisely, the first three properties above assert the mapping α :
727:
1789:
1921:
1770:
1709:
1717:
Michor, P. W. (1989), "Knit products of graded Lie algebras and groups",
308:. This shows that every soluble group is a Zappa–Szép product of a Hall
263:
67:
1912:
1865:
1731:
1156:
1030:, define a multiplication and an inversion mapping by, respectively,
1719:
Proceedings of the Winter School on
Geometry and Physics, Srni
293:
of upper triangular matrices with positive diagonal entries.
1721:, Suppl. Rendiconti Circolo Matematico di Palermo, Ser. II,
1855:
Agore, A.L.; Chirvasitu, A.; Ion, B.; Militaru, G. (2007),
1246:, then the mappings α and β are given by, respectively, α(
845:. This establishes injectivity, and for surjectivity, use
175:
If either (and hence both) of these statements hold, then
336:
to be subgroups of a given group. To motivate this, let
1519:
1499:
1428:
1276:
1566:
1505:
1485:
1414:
1898:Brin, M. G. (2005). "On the Zappa-Szép Product".
1653:Regular Subgroups of Primitive Permutation Groups
444:which turn out to have the following properties:
1750:Proceedings of the National Academy of Sciences
1226:be an internal Zappa–Szép product of subgroups
1194:is, in fact, an internal Zappa–Szép product of
344:be an internal Zappa–Szép product of subgroups
1486:{\displaystyle h_{1}k_{1}h_{2}k_{1}^{-1}\in H}
296:One of the most important examples of this is
8:
1851:; Edizioni Cremonense, Rome, (1942) 119–125.
1655:. American Mathematical Soc. pp. 1–2.
117:. The following statements are equivalent:
1845:: CS1 maint: location missing publisher (
1214:Relation to semidirect and direct products
928:, then the last two properties amount to (
1911:
1864:
1779:
1769:
1730:
1567:{\displaystyle k_{1}h_{2}k_{1}^{-1}\in H}
1549:
1544:
1534:
1524:
1518:
1498:
1468:
1463:
1453:
1443:
1433:
1427:
1403:
1393:
1374:
1369:
1359:
1349:
1339:
1320:
1310:
1294:
1284:
1275:
1857:Factorization problems for finite groups
1019:satisfying the properties above. On the
1640:
1838:
1651:; Cheryl E. Praeger; Jan Saxl (2010).
1833:Atti Secondo Congresso Un. Mat. Ital.
1578:is an internal semidirect product of
7:
300:'s 1937 theorem on the existence of
241:asserts that there exists a unique
912:. If we denote the left action by
25:
1613:is an internal direct product of
70:. It is a generalization of the
1682:(in German), Berlin, New York:
1143:is a group called the external
278:is a Zappa–Szép product of the
1409:
1386:
1383:
1332:
1326:
1303:
1300:
1277:
1270:. This is easy to see because
702:. From these, it follows that
1:
983:Turning this around, suppose
62:) describes a way in which a
328:External Zappa–Szép products
90:Internal Zappa–Szép products
66:can be constructed from two
908:on (the underlying set of)
884:on (the underlying set of)
1963:
179:is said to be an internal
1900:Communications in Algebra
1883:10.1007/s10468-009-9145-6
1824:10.1080/00927878108822621
1630:Complement (group theory)
920:and the right action by
151:, there exists a unique
44:Zappa–Rédei–Szép product
250:upper triangular matrix
1798:Acta Sci. Math. Szeged
1568:
1507:
1493:since by normality of
1487:
1416:
312:-subgroup and a Sylow
289:) and the group (say)
1922:10.1081/AGB-200047404
1771:10.1073/pnas.21.7.469
1569:
1508:
1488:
1417:
78:. It is named after
1517:
1497:
1426:
1274:
1186:, respectively, and
991:are groups (and let
757:) is a bijection of
262:entries on the main
213:general linear group
1875:2007math......3471A
1762:1935PNAS...21..469M
1741:1992math......4220M
1557:
1476:
1382:
765:(Indeed, suppose α(
76:semidirect products
56:exact factorization
42:(also known as the
27:Mathematics concept
1564:
1540:
1503:
1483:
1459:
1412:
1365:
1145:Zappa–Szép product
888:and that β :
229:. For each matrix
181:Zappa–Szép product
40:Zappa–Szép product
18:Zappa-Szep product
1693:978-3-540-03825-2
1662:978-0-8218-4654-4
1649:Martin W. Liebeck
1589:If, in addition,
1506:{\displaystyle H}
1021:cartesian product
60:bicrossed product
16:(Redirected from
1954:
1933:
1915:
1893:
1868:
1850:
1844:
1836:
1826:
1805:
1792:
1783:
1773:
1743:
1734:
1712:
1680:Endliche Gruppen
1667:
1666:
1645:
1609:. In this case,
1574:. In this case,
1573:
1571:
1570:
1565:
1556:
1548:
1539:
1538:
1529:
1528:
1512:
1510:
1509:
1504:
1492:
1490:
1489:
1484:
1475:
1467:
1458:
1457:
1448:
1447:
1438:
1437:
1421:
1419:
1418:
1413:
1408:
1407:
1398:
1397:
1381:
1373:
1364:
1363:
1354:
1353:
1344:
1343:
1325:
1324:
1315:
1314:
1299:
1298:
1289:
1288:
416:). This defines
372:, there exist α(
239:QR decomposition
113:be subgroups of
100:identity element
98:be a group with
21:
1962:
1961:
1957:
1956:
1955:
1953:
1952:
1951:
1937:
1936:
1897:
1854:
1837:
1830:
1809:
1795:
1747:
1716:
1694:
1684:Springer-Verlag
1674:
1671:
1670:
1663:
1647:
1646:
1642:
1637:
1627:
1530:
1520:
1515:
1514:
1495:
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1449:
1439:
1429:
1424:
1423:
1399:
1389:
1355:
1345:
1335:
1316:
1306:
1290:
1280:
1272:
1271:
1216:
1103:
1096:
1089:
1082:
1075:
1068:
1061:
1054:
1047:
1040:
979:
973:
966:
960:
953:
947:
940:
934:
844:
837:
822:
807:
793:
786:
775:
697:
690:
679:
672:
657:
646:
639:
628:
622:
612:
605:
594:
583:
577:
563:
556:
545:
539:
521:
514:
503:
497:
330:
227:complex numbers
197:
92:
48:general product
28:
23:
22:
15:
12:
11:
5:
1960:
1958:
1950:
1949:
1939:
1938:
1935:
1934:
1906:(2): 393–424.
1895:
1852:
1828:
1818:(8): 841–882,
1807:
1793:
1756:(7): 469–472,
1745:
1714:
1713:, Kap. VI, §4.
1692:
1669:
1668:
1661:
1639:
1638:
1636:
1633:
1626:
1623:
1563:
1560:
1555:
1552:
1547:
1543:
1537:
1533:
1527:
1523:
1502:
1482:
1479:
1474:
1471:
1466:
1462:
1456:
1452:
1446:
1442:
1436:
1432:
1411:
1406:
1402:
1396:
1392:
1388:
1385:
1380:
1377:
1372:
1368:
1362:
1358:
1352:
1348:
1342:
1338:
1334:
1331:
1328:
1323:
1319:
1313:
1309:
1305:
1302:
1297:
1293:
1287:
1283:
1279:
1215:
1212:
1174:are subgroups
1147:of the groups
1133:
1132:
1105:
1101:
1094:
1087:
1080:
1073:
1066:
1059:
1052:
1045:
1038:
977:
971:
964:
958:
951:
945:
938:
932:
842:
835:
820:
805:
791:
784:
773:
763:
762:
745:, the mapping
735:
714:, the mapping
695:
688:
677:
670:
664:
663:
655:
644:
637:
626:
620:
614:
610:
603:
592:
581:
575:
565:
561:
554:
543:
537:
527:
519:
512:
501:
495:
489:
329:
326:
306:soluble groups
243:unitary matrix
196:
193:
173:
172:
141:
91:
88:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1959:
1948:
1945:
1944:
1942:
1931:
1927:
1923:
1919:
1914:
1909:
1905:
1901:
1896:
1892:
1888:
1884:
1880:
1876:
1872:
1867:
1862:
1858:
1853:
1848:
1842:
1834:
1829:
1825:
1821:
1817:
1813:
1812:Comm. Algebra
1808:
1803:
1799:
1794:
1791:
1787:
1782:
1777:
1772:
1767:
1763:
1759:
1755:
1751:
1746:
1742:
1738:
1733:
1728:
1724:
1720:
1715:
1711:
1707:
1703:
1699:
1695:
1689:
1685:
1681:
1677:
1673:
1672:
1664:
1658:
1654:
1650:
1644:
1641:
1634:
1632:
1631:
1624:
1622:
1620:
1616:
1612:
1608:
1604:
1600:
1596:
1593:is normal in
1592:
1587:
1585:
1581:
1577:
1561:
1558:
1553:
1550:
1545:
1541:
1535:
1531:
1525:
1521:
1500:
1480:
1477:
1472:
1469:
1464:
1460:
1454:
1450:
1444:
1440:
1434:
1430:
1404:
1400:
1394:
1390:
1378:
1375:
1370:
1366:
1360:
1356:
1350:
1346:
1340:
1336:
1329:
1321:
1317:
1311:
1307:
1295:
1291:
1285:
1281:
1269:
1265:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1233:
1229:
1225:
1221:
1213:
1211:
1209:
1205:
1201:
1197:
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1158:
1154:
1150:
1146:
1142:
1138:
1130:
1126:
1122:
1118:
1114:
1110:
1106:
1100:
1093:
1086:
1079:
1072:
1065:
1058:
1051:
1044:
1037:
1033:
1032:
1031:
1029:
1025:
1022:
1018:
1014:
1010:
1007:and β :
1006:
1002:
998:
994:
990:
986:
981:
976:
970:
963:
957:
950:
944:
937:
931:
927:
923:
919:
915:
911:
907:
903:
899:
895:
891:
887:
883:
879:
875:
871:
867:
862:
860:
856:
852:
848:
841:
834:
830:
826:
819:
815:
811:
804:
800:
797:
790:
783:
779:
772:
768:
760:
756:
752:
748:
744:
740:
736:
733:
729:
725:
721:
717:
713:
709:
705:
704:
703:
701:
694:
687:
683:
676:
669:
661:
654:
650:
643:
636:
632:
625:
619:
615:
609:
602:
598:
591:
587:
580:
574:
570:
566:
560:
553:
549:
542:
536:
532:
528:
525:
518:
511:
507:
500:
494:
490:
487:
483:
479:
475:
471:
467:
463:
459:
455:
451:
447:
446:
445:
443:
439:
435:
432:and β :
431:
427:
423:
419:
415:
411:
407:
403:
399:
395:
391:
387:
383:
379:
375:
371:
367:
363:
359:
355:
352:of the group
351:
347:
343:
339:
335:
327:
325:
322:
321:George Miller
317:
315:
311:
307:
303:
302:Sylow systems
299:
294:
292:
288:
284:
281:
280:unitary group
277:
273:
269:
265:
261:
258:
254:
251:
248:and a unique
247:
244:
240:
236:
232:
228:
224:
221:
218:
214:
210:
206:
202:
194:
192:
190:
186:
182:
178:
170:
166:
162:
159:and a unique
158:
154:
150:
146:
142:
139:
135:
131:
127:
123:
120:
119:
118:
116:
112:
108:
104:
101:
97:
89:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
34:, especially
33:
19:
1947:Group theory
1913:math/0406044
1903:
1899:
1866:math/0703471
1856:
1832:
1815:
1811:
1801:
1797:
1753:
1749:
1732:math/9204220
1722:
1718:
1679:
1652:
1643:
1628:
1618:
1614:
1610:
1606:
1602:
1598:
1594:
1590:
1588:
1583:
1579:
1575:
1267:
1263:
1259:
1255:
1251:
1247:
1243:
1235:
1231:
1227:
1223:
1219:
1217:
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1171:
1167:
1163:
1159:
1152:
1148:
1144:
1140:
1136:
1134:
1128:
1124:
1120:
1116:
1112:
1108:
1098:
1091:
1084:
1077:
1070:
1063:
1056:
1049:
1042:
1035:
1027:
1023:
1016:
1012:
1008:
1004:
1000:
996:
992:
988:
984:
982:
974:
968:
961:
955:
948:
942:
935:
929:
925:
921:
917:
913:
909:
905:
902:right action
897:
893:
889:
885:
881:
873:
869:
865:
863:
858:
854:
850:
846:
839:
832:
828:
824:
817:
813:
809:
802:
798:
795:
788:
781:
777:
770:
766:
764:
758:
754:
750:
746:
742:
738:
731:
723:
719:
715:
711:
707:
699:
692:
685:
681:
674:
667:
665:
659:
652:
648:
641:
634:
630:
623:
617:
607:
600:
596:
595:) α(β(
589:
585:
578:
572:
568:
558:
551:
547:
540:
534:
530:
523:
516:
509:
505:
498:
492:
485:
481:
477:
473:
469:
465:
461:
457:
453:
449:
441:
437:
433:
429:
425:
421:
413:
409:
405:
401:
397:
393:
389:
385:
381:
377:
373:
369:
365:
361:
357:
353:
349:
345:
341:
337:
333:
331:
318:
313:
309:
295:
290:
286:
282:
275:
271:
267:
252:
245:
234:
230:
219:
208:
204:
200:
198:
188:
184:
180:
176:
174:
168:
164:
160:
156:
152:
148:
144:
137:
133:
129:
125:
121:
114:
110:
106:
102:
95:
93:
59:
55:
52:knit product
51:
47:
43:
39:
36:group theory
29:
1725:: 171–175,
1676:Huppert, B.
878:left action
651:)) β(
356:. For each
298:Philip Hall
82:(1940) and
80:Guido Zappa
32:mathematics
1635:References
1176:isomorphic
396:such that
266:such that
217:invertible
167:such that
105:, and let
1835:, Bologna
1597:, then α(
1559:∈
1551:−
1478:∈
1470:−
1376:−
1069: α(
1048:) (
737:For each
728:bijection
706:For each
420:α :
364:and each
319:In 1935,
225:over the
143:For each
84:Jenő Szép
68:subgroups
1941:Category
1930:15169734
1891:18024087
1841:citation
1790:16588002
1678:(1967),
1625:See also
1127:,
1119:,
1097:)
1090:,
1076:,
857:,
831:,
816:,
787:). Then
666:for all
658:,
647:,
599:,
588:,
546:) = β(β(
522:,
472:for all
418:mappings
334:a priori
264:diagonal
257:positive
223:matrices
195:Examples
1871:Bibcode
1804:: 57–61
1781:1076628
1758:Bibcode
1737:Bibcode
1702:0224703
1202:} and {
1166:} and {
1157:subsets
1115:) = (α(
823:)) = α(
726:) is a
274:. Thus
211:), the
1928:
1889:
1788:
1778:
1710:527050
1708:
1700:
1690:
1659:
1258:and β(
1240:normal
1155:. The
808:) = α(
776:) = α(
633:) = β(
584:) = α(
508:) = α(
460:and β(
384:and β(
237:, the
169:g = hk
72:direct
38:, the
1926:S2CID
1908:arXiv
1887:S2CID
1861:arXiv
1727:arXiv
1422:and
1256:k h k
1234:. If
1135:Then
1123:), β(
1083:), β(
1062:) = (
954:and (
900:is a
876:is a
861:)).)
838:)) =
392:) in
380:) in
255:with
220:n × n
203:= GL(
64:group
1847:link
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