3140:
3108:
3071:
3232:
3208:
3256:
3180:
2951:
3381:
3365:
3347:
3160:
3128:
3096:
3323:
3313:
3303:
3155:
3150:
3123:
3118:
3091:
3034:
3291:
3086:
2939:
3442:
3426:
3247:
3223:
3050:
3045:
3458:
3271:
3055:
3408:
3199:
3040:
3018:
2918:
1907:
59:
2525:
1523:
1995:
1949:
1756:
1629:
2302:
1593:
1440:
2194:
3588:
3556:
1835:
1552:
504:
479:
442:
2409:. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if
806:
3139:
3107:
3070:
2950:
827:, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
4117:
3973:
364:
3904:
3809:
3231:
3207:
3255:
2938:
2221:
2048:
314:
3179:
3380:
3364:
3346:
3159:
3127:
3095:
1144:
is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element
799:
309:
3322:
3312:
3302:
3154:
3149:
3122:
3117:
3090:
3033:
2452:
3872:
3853:
3831:
1492:
1444:
under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of
3290:
3085:
3441:
3425:
3246:
3222:
3049:
3044:
3457:
3270:
3054:
1962:
1916:
3407:
3198:
3039:
1726:
725:
1556:
because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the
792:
4464:
2380:
They are also the equivalence classes for a suitable equivalence relation and their number is equal to .
409:
223:
4433:
3966:
1602:
141:
3497:
2238:
1569:
1416:
3658:
3596:
3187:
3077:
2154:
1662:
1384:
873:
31:
17:
4028:
3568:
3536:
1815:
1532:
607:
341:
218:
106:
487:
462:
425:
4428:
2995:
1129:, but it is more natural and usually just as easy to test the two closure conditions separately.
4313:
757:
547:
4459:
4299:
4251:
3959:
3017:
1058:
631:
4275:
4269:
4033:
3819:
3653:
3626:
2966:
2917:
2324:
2129:
1855:
571:
559:
177:
111:
8:
4008:
3982:
2207:
831:
146:
41:
3942:
4208:
4024:
4018:
2531:
1456:
131:
103:
4131:
3910:
3900:
3878:
3868:
3849:
3827:
3805:
2991:
2858:, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.
1952:
1761:
1718:
536:
379:
273:
4370:
702:
4438:
4375:
4350:
4342:
4334:
4326:
4318:
4305:
4287:
4281:
3929:
3648:
3630:
3294:
1851:
1241:
838:
687:
679:
671:
663:
655:
643:
583:
523:
513:
355:
297:
172:
3592:
the sum of two even integers is even, and the negative of an even integer is even.
1870:
the set-theoretic union of the subgroups, not the set-theoretic union itself.) If
4391:
4225:
4094:
4003:
3797:
3663:
3643:
3611:
3330:
3183:
3001:
2876:
2406:
1290:
771:
764:
750:
707:
595:
518:
348:
262:
202:
82:
4396:
4238:
4191:
4184:
4177:
4170:
4142:
4109:
4079:
4074:
4064:
4013:
3923:
3890:
3074:
2147:
1051:
778:
714:
404:
384:
321:
286:
207:
197:
182:
167:
121:
98:
3933:
3757:
4453:
4406:
4365:
4293:
4214:
4199:
4147:
4099:
4054:
3914:
3882:
3841:
3622:
3519:
2962:
2027:
1895:
1887:
697:
619:
453:
326:
192:
1906:
4360:
4160:
4122:
4069:
4059:
4049:
3990:
3615:
3561:
3411:
3350:
2892:
2855:
2535:
2437:
1377:
1293:
of an element in a subgroup is the inverse of the element in the group: if
970:". Some authors also exclude the trivial group from being proper (that is,
820:
552:
251:
240:
187:
162:
157:
116:
87:
50:
3894:
4089:
4084:
2880:
824:
4401:
719:
447:
4261:
2117:
1012:
540:
2128:; the left cosets are the equivalence classes corresponding to the
1863:
3716:
3714:
2344:
1859:
419:
333:
1107:. These two conditions can be combined into one, that for every
3711:
935:
864:
also forms a group under the operation ∗. More precisely,
842:
58:
2335:, respectively. In particular, the order of every subgroup of
2068:
2044:
1174:
If the group operation is instead denoted by addition, then
1015:, but this article will only deal with subgroups of groups.
3899:(8th ed.). Boston, MA: Brooks/Cole Cengage Learning.
3951:
2413:
is the lowest prime dividing the order of a finite group
1647:, namely the intersection of all of subgroups containing
3763:
3687:
2034:
into equal-size, non-overlapping sets. The index is 4.
1039:
is written multiplicatively, denoted by juxtaposition.
3491:}. These are the permutations that have only 2-cycles:
3726:
3571:
3539:
2455:
2241:
2157:
1965:
1919:
1818:
1729:
1682:
if and only if it is a finite product of elements of
1605:
1572:
1535:
1495:
1419:
490:
465:
428:
3775:
3675:
2885:
Below are all its subgroups, ordered by cardinality.
3947:. Department of Mathematics University of Illinois.
3921:
3720:
1643:, then there exists a smallest subgroup containing
3699:
3582:
3550:
2519:
2296:
2188:
2026:(written using additive notation since this is an
1989:
1943:
1862:here is the usual set-theoretic intersection, the
1829:
1750:
1623:
1587:
1546:
1517:
1434:
498:
473:
436:
2846:is the top-left quadrant of the Cayley table for
2838:is the top-left quadrant of the Cayley table for
4451:
2520:{\displaystyle G=\left\{0,4,2,6,1,5,3,7\right\}}
2038:
1234:
3618:is a subgroup of the additive group of vectors.
1518:{\displaystyle 2\mathbb {Z} \cup 3\mathbb {Z} }
1244:of a subgroup is the identity of the group: if
1035:. For now, assume that the group operation of
1007:The same definitions apply more generally when
3738:
3967:
3506:And 3 permutations with two 2-cycles.
952:). This is often represented notationally by
800:
3863:Dummit, David S.; Foote, Richard M. (2004).
2124:is contained in precisely one left coset of
2030:). Together they partition the entire group
1959:contains only 0 and 4, and is isomorphic to
3922:Kurzweil, Hans; Stellmacher, Bernd (1998).
3862:
3769:
2861:
2424:
1990:{\displaystyle \mathbb {Z} /2\mathbb {Z} .}
1944:{\displaystyle \mathbb {Z} /8\mathbb {Z} ,}
1778:is the smallest positive integer for which
919:} consisting of just the identity element.
27:Subset of a group that forms a group itself
3974:
3960:
3818:
3693:
1751:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
807:
793:
3944:Abstract Algebra: The Basic Graduate Year
3573:
3544:
2806:This group has two nontrivial subgroups:
1980:
1967:
1934:
1921:
1820:
1744:
1731:
1608:
1575:
1537:
1511:
1500:
1422:
492:
467:
430:
3796:
3732:
3456:
3440:
3424:
3406:
3379:
3363:
3345:
3321:
3311:
3301:
3289:
3254:
3230:
3206:
3178:
3138:
3106:
3069:
3016:
2916:
1905:
1850:The subgroups of any given group form a
1182:, which is the condition that for every
3889:
3781:
3681:
3603:is a subgroup of the additive group of
3000:It is one of the two nontrivial proper
1403:. For example, the intersection of the
1212:should be edited to say that for every
14:
4452:
4118:Classification of finite simple groups
3013:(The other one is its Klein subgroup.)
1866:of a set of subgroups is the subgroup
1686:and their inverses, possibly repeated.
1152:generates a finite cyclic subgroup of
365:Classification of finite simple groups
3955:
3840:
3804:, vol. 1 (2nd ed.), Dover,
3705:
2889:(except those of cardinality 1 and 2)
3867:(3rd ed.). Hoboken, NJ: Wiley.
3940:
3744:
3522:is the unique subgroup of order 1.
2339:(and the order of every element of
24:
3531:The even integers form a subgroup
25:
4476:
3826:(1st ed.), Springer-Verlag,
3525:
2879:whose elements correspond to the
2049:Lagrange's theorem (group theory)
1624:{\displaystyle \mathbb {R} ^{2}.}
1018:
3508:(white background, bold numbers)
3484:of order 2 generates a subgroup
3269:
3245:
3221:
3197:
3158:
3153:
3148:
3126:
3121:
3116:
3094:
3089:
3084:
3053:
3048:
3043:
3038:
3032:
2949:
2937:
2297:{\displaystyle ={|G| \over |H|}}
2120:. Furthermore, every element of
1588:{\displaystyle \mathbb {R} ^{2}}
1435:{\displaystyle \mathbb {R} ^{2}}
57:
3848:(2nd ed.), Prentice Hall,
3721:Kurzweil & Stellmacher 1998
2224:states that for a finite group
2202:. The number of left cosets of
2189:{\displaystyle a_{1}^{-1}a_{2}}
3750:
2985:
2898:
2808:
2287:
2279:
2272:
2264:
2254:
2242:
1997:There are four left cosets of
1140:, the test can be simplified:
915:of any group is the subgroup {
726:Infinite dimensional Lie group
13:
1:
3925:Theorie der endlichen Gruppen
3896:Contemporary abstract algebra
3790:
3583:{\displaystyle \mathbb {Z} :}
3551:{\displaystyle 2\mathbb {Z} }
3475:
3398:
3281:
3170:
3061:
2530:and whose group operation is
2417:, then any subgroup of index
2039:Cosets and Lagrange's theorem
1955:under addition. The subgroup
1830:{\displaystyle \mathbb {Z} ,}
1547:{\displaystyle \mathbb {Z} ,}
1467:is a subgroup if and only if
1235:Basic properties of subgroups
1061:under products and inverses.
3758:didactic proof in this video
3513:
2421:(if such exists) is normal.
1854:under inclusion, called the
1770:) for some positive integer
1697:generates a cyclic subgroup
499:{\displaystyle \mathbb {Z} }
474:{\displaystyle \mathbb {Z} }
437:{\displaystyle \mathbb {Z} }
7:
4385:Infinite dimensional groups
3637:
3629:form a subgroup called the
2818:
224:List of group theory topics
10:
4481:
3981:
2042:
1160:, and then the inverse of
29:
4424:
4384:
4260:
4108:
4042:
3989:
3934:10.1007/978-3-642-58816-7
3659:Fully normalized subgroup
3625:, the elements of finite
3500:with one 2-cycle.
3143:Dihedral group of order 8
3111:Dihedral group of order 8
2910:is a subgroup of itself.
2361:are defined analogously:
1878:, then the trivial group
1358:, then the inclusion map
1297:is a subgroup of a group
1248:is a group with identity
32:Subgroup (disambiguation)
4288:Special orthogonal group
3669:
2834:. The Cayley table for
966:is a proper subgroup of
890:. This is often denoted
886:is a group operation on
342:Elementary abelian group
219:Glossary of group theory
3941:Ash, Robert B. (2002).
3770:Dummit & Foote 2004
2862:Example: Subgroups of S
2842:; The Cayley table for
2425:Example: Subgroups of Z
2094:is invertible, the map
1399:is again a subgroup of
996:is sometimes called an
4314:Exceptional Lie groups
3584:
3552:
3468:
3452:
3436:
3420:
3391:
3375:
3359:
3335:
3317:
3307:
3297:
3274:
3250:
3226:
3202:
3163:
3131:
3099:
3058:
2928:
2891:is represented by its
2830:is also a subgroup of
2521:
2298:
2190:
2035:
1991:
1945:
1898:subgroup is the group
1831:
1752:
1663:subgroup generated by
1625:
1589:
1548:
1519:
1436:
1178:should be replaced by
758:Linear algebraic group
500:
475:
438:
4300:Special unitary group
3928:. Springer-Lehrbuch.
3585:
3553:
3460:
3444:
3428:
3410:
3383:
3367:
3349:
3325:
3315:
3305:
3293:
3258:
3234:
3210:
3182:
3142:
3110:
3073:
3020:
2920:
2522:
2299:
2191:
1992:
1946:
1909:
1832:
1753:
1626:
1597:is not a subgroup of
1590:
1549:
1527:is not a subgroup of
1520:
1437:
1368:sending each element
1210:closed under inverses
1180:closed under addition
1176:closed under products
1089:means that for every
1087:Closed under inverses
1065:means that for every
1063:Closed under products
501:
476:
439:
4397:Diffeomorphism group
4276:Special linear group
4270:General linear group
3654:Fixed-point subgroup
3569:
3537:
2967:lattice of subgroups
2453:
2239:
2218:and is denoted by .
2155:
2130:equivalence relation
1963:
1917:
1856:lattice of subgroups
1816:
1727:
1651:; it is denoted by
1603:
1570:
1533:
1493:
1417:
488:
463:
426:
30:For other uses, see
4465:Subgroup properties
4222:Other finite groups
4009:Commutator subgroup
2446:whose elements are
2175:
1874:is the identity of
132:Group homomorphisms
42:Algebraic structure
4252:Rubik's Cube group
4209:Baby monster group
4019:Group homomorphism
3891:Gallian, Joseph A.
3820:Hungerford, Thomas
3580:
3548:
3502:(green background)
3469:
3453:
3437:
3421:
3392:
3376:
3360:
3336:
3318:
3308:
3298:
3275:
3251:
3227:
3203:
3164:
3132:
3100:
3059:
3021:Alternating group
2994:contains only the
2929:
2517:
2294:
2222:Lagrange's theorem
2186:
2158:
2036:
1987:
1941:
1827:
1748:
1661:and is called the
1621:
1585:
1544:
1515:
1487:. A non-example:
1432:
830:Formally, given a
608:Special orthogonal
496:
471:
434:
315:Lagrange's theorem
4447:
4446:
4132:Alternating group
3906:978-1-133-59970-8
3811:978-0-486-47189-1
3509:
3503:
3480:Each permutation
3473:
3472:
3396:
3395:
3340:
3339:
3334:
3279:
3278:
3168:
3167:
3014:
2996:even permutations
2992:alternating group
2983:
2982:
2903:Like each group,
2890:
2804:
2803:
2532:addition modulo 8
2292:
2053:Given a subgroup
1810:is isomorphic to
1448:is a subgroup of
1354:is a subgroup of
1259:is a subgroup of
1047:is a subgroup of
988:is a subgroup of
904:is a subgroup of
868:is a subgroup of
817:
816:
392:
391:
274:Alternating group
231:
230:
16:(Redirected from
4472:
4439:Abstract algebra
4376:Quaternion group
4306:Symplectic group
4282:Orthogonal group
3976:
3969:
3962:
3953:
3952:
3948:
3937:
3918:
3886:
3865:Abstract algebra
3858:
3836:
3814:
3798:Jacobson, Nathan
3785:
3779:
3773:
3767:
3761:
3754:
3748:
3742:
3736:
3730:
3724:
3718:
3709:
3703:
3697:
3691:
3685:
3679:
3649:Fitting subgroup
3631:torsion subgroup
3606:
3602:
3591:
3589:
3587:
3586:
3581:
3576:
3559:
3557:
3555:
3554:
3549:
3547:
3520:trivial subgroup
3507:
3501:
3496:There are the 6
3490:
3483:
3467:
3451:
3435:
3419:
3403:
3402:
3390:
3374:
3358:
3342:
3341:
3328:
3326:Klein four-group
3316:Klein four-group
3306:Klein four-group
3295:Klein four-group
3286:
3285:
3273:
3265:
3259:Symmetric group
3249:
3241:
3235:Symmetric group
3225:
3217:
3211:Symmetric group
3201:
3193:
3175:
3174:
3162:
3157:
3152:
3130:
3125:
3120:
3098:
3093:
3088:
3066:
3065:
3057:
3052:
3047:
3042:
3036:
3027:
3012:
3010:
3002:normal subgroups
2975:
2953:
2944:All 30 subgroups
2941:
2927:
2921:Symmetric group
2913:
2912:
2909:
2888:
2874:
2853:
2849:
2845:
2841:
2837:
2833:
2829:
2825:
2820:
2815:
2810:
2541:
2540:
2526:
2524:
2523:
2518:
2516:
2512:
2445:
2435:
2420:
2416:
2412:
2405:is said to be a
2404:
2400:
2396:
2392:
2379:
2354:
2352:
2342:
2338:
2334:
2330:
2322:
2320:
2314:
2312:
2303:
2301:
2300:
2295:
2293:
2291:
2290:
2282:
2276:
2275:
2267:
2261:
2231:
2227:
2217:
2213:
2205:
2201:
2197:
2195:
2193:
2192:
2187:
2185:
2184:
2174:
2166:
2146:
2127:
2123:
2115:
2104:
2093:
2089:
2065:, we define the
2064:
2060:
2056:
2033:
2025:
2018:
2011:
2004:
2000:
1996:
1994:
1993:
1988:
1983:
1975:
1970:
1958:
1950:
1948:
1947:
1942:
1937:
1929:
1924:
1912:
1901:
1893:
1885:
1877:
1873:
1852:complete lattice
1843:is said to have
1842:
1838:
1836:
1834:
1833:
1828:
1823:
1809:
1808:
1799:
1791:
1787:
1777:
1773:
1768:
1759:
1757:
1755:
1754:
1749:
1747:
1739:
1734:
1716:
1715:
1706:
1705:
1696:
1692:
1685:
1681:
1680:
1671:
1668:. An element of
1666:
1660:
1659:
1650:
1646:
1642:
1638:
1632:
1630:
1628:
1627:
1622:
1617:
1616:
1611:
1596:
1594:
1592:
1591:
1586:
1584:
1583:
1578:
1563:
1559:
1555:
1553:
1551:
1550:
1545:
1540:
1526:
1524:
1522:
1521:
1516:
1514:
1503:
1486:
1476:
1466:
1462:
1451:
1447:
1443:
1441:
1439:
1438:
1433:
1431:
1430:
1425:
1410:
1406:
1402:
1398:
1394:
1390:
1375:
1371:
1367:
1357:
1353:
1346:
1329:
1312:
1309:are elements of
1308:
1304:
1300:
1296:
1285:
1269:
1262:
1258:
1254:
1247:
1230:
1226:
1219:
1215:
1207:
1203:
1193:
1189:
1185:
1169:
1163:
1159:
1155:
1151:
1147:
1143:
1135:
1128:
1124:
1118:
1114:
1110:
1106:
1102:
1096:
1092:
1084:
1080:
1076:
1072:
1068:
1057:is nonempty and
1056:
1050:
1046:
1038:
1034:
1030:
1027:is a group, and
1026:
1011:is an arbitrary
1010:
1003:
995:
991:
987:
980:
969:
965:
961:
951:
941:
933:
929:
913:trivial subgroup
907:
903:
899:
889:
885:
871:
867:
863:
859:
851:
847:
839:binary operation
836:
809:
802:
795:
751:Algebraic groups
524:Hyperbolic group
514:Arithmetic group
505:
503:
502:
497:
495:
480:
478:
477:
472:
470:
443:
441:
440:
435:
433:
356:Schur multiplier
310:Cauchy's theorem
298:Quaternion group
246:
245:
72:
71:
61:
48:
37:
36:
21:
4480:
4479:
4475:
4474:
4473:
4471:
4470:
4469:
4450:
4449:
4448:
4443:
4420:
4392:Conformal group
4380:
4354:
4346:
4338:
4330:
4322:
4256:
4248:
4235:
4226:Symmetric group
4205:
4195:
4188:
4181:
4174:
4166:
4157:
4153:
4143:Sporadic groups
4137:
4128:
4110:Discrete groups
4104:
4095:Wallpaper group
4075:Solvable groups
4043:Types of groups
4038:
4004:Normal subgroup
3985:
3980:
3907:
3875:
3856:
3834:
3812:
3793:
3788:
3780:
3776:
3768:
3764:
3755:
3751:
3743:
3739:
3731:
3727:
3719:
3712:
3704:
3700:
3694:Hungerford 1974
3692:
3688:
3680:
3676:
3672:
3664:Stable subgroup
3644:Cartan subgroup
3640:
3612:linear subspace
3604:
3600:
3572:
3570:
3567:
3566:
3564:
3543:
3538:
3535:
3534:
3532:
3528:
3516:
3492:
3485:
3481:
3478:
3466:
3462:
3450:
3446:
3434:
3430:
3418:
3414:
3401:
3389:
3385:
3373:
3369:
3357:
3353:
3331:normal subgroup
3327:
3284:
3267:
3266:
3264:
3260:
3243:
3242:
3240:
3236:
3219:
3218:
3216:
3212:
3195:
3194:
3191:
3186:
3184:Symmetric group
3173:
3147:
3145:
3144:
3115:
3113:
3112:
3083:
3081:
3080:
3064:
3037:
3031:
3029:
3028:
3026:
3022:
3009:
3005:
2999:
2988:
2979:
2978:
2977:
2976:
2974:
2970:
2959:
2958:
2957:
2954:
2946:
2945:
2942:
2926:
2922:
2908:
2904:
2901:
2886:
2884:
2877:symmetric group
2873:
2869:
2867:
2865:
2851:
2847:
2843:
2839:
2835:
2831:
2827:
2824:= {0, 4, 2, 6}
2817:
2807:
2466:
2462:
2454:
2451:
2450:
2444:
2440:
2433:
2430:
2428:
2418:
2414:
2410:
2407:normal subgroup
2402:
2398:
2394:
2384:
2362:
2350:
2348:
2340:
2336:
2332:
2328:
2318:
2316:
2310:
2308:
2286:
2278:
2277:
2271:
2263:
2262:
2260:
2240:
2237:
2236:
2229:
2228:and a subgroup
2225:
2215:
2211:
2203:
2199:
2180:
2176:
2167:
2162:
2156:
2153:
2152:
2150:
2145:
2138:
2132:
2125:
2121:
2106:
2095:
2091:
2072:
2062:
2058:
2054:
2051:
2043:Main articles:
2041:
2031:
2020:
2013:
2006:
2002:
1998:
1979:
1971:
1966:
1964:
1961:
1960:
1956:
1933:
1925:
1920:
1918:
1915:
1914:
1910:
1899:
1891:
1879:
1875:
1871:
1840:
1819:
1817:
1814:
1813:
1811:
1802:
1801:
1797:
1789:
1779:
1775:
1771:
1763:
1743:
1735:
1730:
1728:
1725:
1724:
1722:
1709:
1708:
1699:
1698:
1694:
1690:
1683:
1674:
1673:
1669:
1664:
1653:
1652:
1648:
1644:
1640:
1639:is a subset of
1636:
1612:
1607:
1606:
1604:
1601:
1600:
1598:
1579:
1574:
1573:
1571:
1568:
1567:
1565:
1561:
1557:
1536:
1534:
1531:
1530:
1528:
1510:
1499:
1494:
1491:
1490:
1488:
1478:
1468:
1464:
1460:
1449:
1445:
1426:
1421:
1420:
1418:
1415:
1414:
1412:
1408:
1404:
1400:
1396:
1392:
1388:
1376:to itself is a
1373:
1369:
1359:
1355:
1351:
1344:
1331:
1327:
1314:
1310:
1306:
1302:
1298:
1294:
1283:
1276:
1271:
1268:
1264:
1260:
1256:
1253:
1249:
1245:
1237:
1228:
1221:
1217:
1213:
1205:
1195:
1191:
1187:
1183:
1165:
1161:
1157:
1156:, say of order
1153:
1149:
1145:
1141:
1133:
1126:
1120:
1116:
1112:
1108:
1104:
1098:
1094:
1090:
1082:
1078:
1074:
1070:
1066:
1054:
1048:
1044:
1036:
1032:
1031:is a subset of
1028:
1024:
1021:
1008:
1001:
993:
989:
985:
971:
967:
963:
953:
943:
939:
931:
927:
924:proper subgroup
905:
901:
891:
887:
877:
869:
865:
861:
857:
849:
845:
834:
813:
784:
783:
772:Abelian variety
765:Reductive group
753:
743:
742:
741:
740:
691:
683:
675:
667:
659:
632:Special unitary
543:
529:
528:
510:
509:
491:
489:
486:
485:
466:
464:
461:
460:
429:
427:
424:
423:
415:
414:
405:Discrete groups
394:
393:
349:Frobenius group
294:
281:
270:
263:Symmetric group
259:
243:
233:
232:
83:Normal subgroup
69:
49:
40:
35:
28:
23:
22:
15:
12:
11:
5:
4478:
4468:
4467:
4462:
4445:
4444:
4442:
4441:
4436:
4431:
4425:
4422:
4421:
4419:
4418:
4415:
4412:
4409:
4404:
4399:
4394:
4388:
4386:
4382:
4381:
4379:
4378:
4373:
4371:Poincaré group
4368:
4363:
4357:
4356:
4352:
4348:
4344:
4340:
4336:
4332:
4328:
4324:
4320:
4316:
4310:
4309:
4303:
4297:
4291:
4285:
4279:
4273:
4266:
4264:
4258:
4257:
4255:
4254:
4249:
4244:
4239:Dihedral group
4236:
4231:
4223:
4219:
4218:
4212:
4206:
4203:
4197:
4193:
4186:
4179:
4172:
4167:
4164:
4158:
4155:
4151:
4145:
4139:
4138:
4135:
4129:
4126:
4120:
4114:
4112:
4106:
4105:
4103:
4102:
4097:
4092:
4087:
4082:
4080:Symmetry group
4077:
4072:
4067:
4065:Infinite group
4062:
4057:
4055:Abelian groups
4052:
4046:
4044:
4040:
4039:
4037:
4036:
4031:
4029:direct product
4021:
4016:
4014:Quotient group
4011:
4006:
4001:
3995:
3993:
3987:
3986:
3979:
3978:
3971:
3964:
3956:
3950:
3949:
3938:
3919:
3905:
3887:
3873:
3860:
3854:
3842:Artin, Michael
3838:
3832:
3816:
3810:
3792:
3789:
3787:
3786:
3774:
3762:
3749:
3737:
3725:
3710:
3698:
3686:
3673:
3671:
3668:
3667:
3666:
3661:
3656:
3651:
3646:
3639:
3636:
3635:
3634:
3619:
3608:
3593:
3579:
3575:
3546:
3542:
3527:
3526:Other examples
3524:
3515:
3512:
3511:
3510:
3504:
3498:transpositions
3477:
3474:
3471:
3470:
3464:
3454:
3448:
3438:
3432:
3422:
3416:
3400:
3397:
3394:
3393:
3387:
3377:
3371:
3361:
3355:
3338:
3337:
3319:
3309:
3299:
3283:
3280:
3277:
3276:
3262:
3252:
3238:
3228:
3214:
3204:
3189:
3172:
3169:
3166:
3165:
3136:
3133:
3104:
3101:
3075:Dihedral group
3063:
3060:
3024:
3007:
2987:
2984:
2981:
2980:
2972:
2963:Hasse diagrams
2961:
2960:
2955:
2948:
2947:
2943:
2936:
2935:
2934:
2933:
2932:
2930:
2924:
2906:
2900:
2897:
2883:of 4 elements.
2871:
2866:
2863:
2860:
2802:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2773:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2744:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2715:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2686:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2657:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2628:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2599:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2570:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2528:
2527:
2515:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2465:
2461:
2458:
2442:
2429:
2426:
2423:
2305:
2304:
2289:
2285:
2281:
2274:
2270:
2266:
2259:
2256:
2253:
2250:
2247:
2244:
2206:is called the
2183:
2179:
2173:
2170:
2165:
2161:
2148:if and only if
2143:
2136:
2040:
2037:
2028:additive group
1986:
1982:
1978:
1974:
1969:
1953:integers mod 8
1940:
1936:
1932:
1928:
1923:
1904:
1903:
1848:
1845:infinite order
1826:
1822:
1792:is called the
1746:
1742:
1738:
1733:
1689:Every element
1687:
1633:
1620:
1615:
1610:
1582:
1577:
1560:-axis and the
1543:
1539:
1513:
1509:
1506:
1502:
1498:
1453:
1429:
1424:
1381:
1348:
1342:
1325:
1287:
1281:
1274:
1266:
1263:with identity
1251:
1236:
1233:
1220:, the inverse
1172:
1171:
1130:
1119:, the element
1097:, the inverse
1077:, the product
1052:if and only if
1020:
1019:Subgroup tests
1017:
930:is a subgroup
823:, a branch of
815:
814:
812:
811:
804:
797:
789:
786:
785:
782:
781:
779:Elliptic curve
775:
774:
768:
767:
761:
760:
754:
749:
748:
745:
744:
739:
738:
735:
732:
728:
724:
723:
722:
717:
715:Diffeomorphism
711:
710:
705:
700:
694:
693:
689:
685:
681:
677:
673:
669:
665:
661:
657:
652:
651:
640:
639:
628:
627:
616:
615:
604:
603:
592:
591:
580:
579:
572:Special linear
568:
567:
560:General linear
556:
555:
550:
544:
535:
534:
531:
530:
527:
526:
521:
516:
508:
507:
494:
482:
469:
456:
454:Modular groups
452:
451:
450:
445:
432:
416:
413:
412:
407:
401:
400:
399:
396:
395:
390:
389:
388:
387:
382:
377:
374:
368:
367:
361:
360:
359:
358:
352:
351:
345:
344:
339:
330:
329:
327:Hall's theorem
324:
322:Sylow theorems
318:
317:
312:
304:
303:
302:
301:
295:
290:
287:Dihedral group
283:
282:
277:
271:
266:
260:
255:
244:
239:
238:
235:
234:
229:
228:
227:
226:
221:
213:
212:
211:
210:
205:
200:
195:
190:
185:
180:
178:multiplicative
175:
170:
165:
160:
152:
151:
150:
149:
144:
136:
135:
127:
126:
125:
124:
122:Wreath product
119:
114:
109:
107:direct product
101:
99:Quotient group
93:
92:
91:
90:
85:
80:
70:
67:
66:
63:
62:
54:
53:
26:
9:
6:
4:
3:
2:
4477:
4466:
4463:
4461:
4458:
4457:
4455:
4440:
4437:
4435:
4432:
4430:
4427:
4426:
4423:
4416:
4413:
4410:
4408:
4407:Quantum group
4405:
4403:
4400:
4398:
4395:
4393:
4390:
4389:
4387:
4383:
4377:
4374:
4372:
4369:
4367:
4366:Lorentz group
4364:
4362:
4359:
4358:
4355:
4349:
4347:
4341:
4339:
4333:
4331:
4325:
4323:
4317:
4315:
4312:
4311:
4307:
4304:
4301:
4298:
4295:
4294:Unitary group
4292:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4267:
4265:
4263:
4259:
4253:
4250:
4247:
4243:
4240:
4237:
4234:
4230:
4227:
4224:
4221:
4220:
4216:
4215:Monster group
4213:
4210:
4207:
4201:
4200:Fischer group
4198:
4196:
4189:
4182:
4175:
4169:Janko groups
4168:
4162:
4159:
4149:
4148:Mathieu group
4146:
4144:
4141:
4140:
4133:
4130:
4124:
4121:
4119:
4116:
4115:
4113:
4111:
4107:
4101:
4100:Trivial group
4098:
4096:
4093:
4091:
4088:
4086:
4083:
4081:
4078:
4076:
4073:
4071:
4070:Simple groups
4068:
4066:
4063:
4061:
4060:Cyclic groups
4058:
4056:
4053:
4051:
4050:Finite groups
4048:
4047:
4045:
4041:
4035:
4032:
4030:
4026:
4022:
4020:
4017:
4015:
4012:
4010:
4007:
4005:
4002:
4000:
3997:
3996:
3994:
3992:
3991:Basic notions
3988:
3984:
3977:
3972:
3970:
3965:
3963:
3958:
3957:
3954:
3946:
3945:
3939:
3935:
3931:
3927:
3926:
3920:
3916:
3912:
3908:
3902:
3898:
3897:
3892:
3888:
3884:
3880:
3876:
3874:9780471452348
3870:
3866:
3861:
3857:
3855:9780132413770
3851:
3847:
3843:
3839:
3835:
3833:9780387905181
3829:
3825:
3821:
3817:
3813:
3807:
3803:
3802:Basic algebra
3799:
3795:
3794:
3784:, p. 81.
3783:
3778:
3772:, p. 90.
3771:
3766:
3759:
3753:
3746:
3741:
3735:, p. 41.
3734:
3733:Jacobson 2009
3729:
3722:
3717:
3715:
3708:, p. 43.
3707:
3702:
3696:, p. 32.
3695:
3690:
3684:, p. 61.
3683:
3678:
3674:
3665:
3662:
3660:
3657:
3655:
3652:
3650:
3647:
3645:
3642:
3641:
3632:
3628:
3624:
3623:abelian group
3620:
3617:
3613:
3609:
3598:
3594:
3577:
3563:
3540:
3530:
3529:
3523:
3521:
3505:
3499:
3495:
3494:
3493:
3489:
3461:Cyclic group
3459:
3455:
3445:Cyclic group
3443:
3439:
3429:Cyclic group
3427:
3423:
3413:
3409:
3405:
3404:
3384:Cyclic group
3382:
3378:
3368:Cyclic group
3366:
3362:
3352:
3348:
3344:
3343:
3332:
3324:
3320:
3314:
3310:
3304:
3300:
3296:
3292:
3288:
3287:
3272:
3257:
3253:
3248:
3233:
3229:
3224:
3209:
3205:
3200:
3192:
3185:
3181:
3177:
3176:
3161:
3156:
3151:
3141:
3137:
3134:
3129:
3124:
3119:
3109:
3105:
3102:
3097:
3092:
3087:
3079:
3076:
3072:
3068:
3067:
3056:
3051:
3046:
3041:
3035:
3019:
3015:
3003:
2997:
2993:
2968:
2964:
2952:
2940:
2931:
2919:
2915:
2914:
2911:
2896:
2894:
2882:
2878:
2859:
2857:
2850:. The group
2823:
2813:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2774:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2745:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2716:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2687:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2658:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2629:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2600:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2571:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2542:
2539:
2537:
2533:
2513:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2463:
2459:
2456:
2449:
2448:
2447:
2439:
2422:
2408:
2391:
2387:
2381:
2377:
2373:
2369:
2365:
2360:
2356:
2346:
2326:
2283:
2268:
2257:
2251:
2248:
2245:
2235:
2234:
2233:
2223:
2219:
2209:
2181:
2177:
2171:
2168:
2163:
2159:
2149:
2142:
2135:
2131:
2119:
2114:
2110:
2103:
2099:
2087:
2083:
2079:
2075:
2071:
2070:
2050:
2046:
2029:
2024:
2017:
2010:
1984:
1976:
1972:
1954:
1938:
1930:
1926:
1913:is the group
1908:
1897:
1889:
1883:
1869:
1865:
1861:
1858:. (While the
1857:
1853:
1849:
1846:
1824:
1806:
1795:
1786:
1782:
1769:
1767:
1762:the integers
1740:
1736:
1720:
1713:
1703:
1688:
1678:
1667:
1657:
1634:
1618:
1613:
1580:
1541:
1507:
1504:
1496:
1485:
1481:
1475:
1471:
1459:of subgroups
1458:
1454:
1427:
1387:of subgroups
1386:
1382:
1379:
1366:
1362:
1349:
1345:
1338:
1334:
1328:
1321:
1317:
1292:
1288:
1284:
1277:
1243:
1239:
1238:
1232:
1225:
1211:
1202:
1198:
1181:
1177:
1168:
1139:
1131:
1123:
1101:
1088:
1064:
1060:
1053:
1042:
1041:
1040:
1023:Suppose that
1016:
1014:
1005:
999:
982:
978:
974:
960:
956:
950:
946:
937:
936:proper subset
925:
920:
918:
914:
909:
898:
894:
884:
880:
875:
855:
844:
840:
833:
828:
826:
822:
810:
805:
803:
798:
796:
791:
790:
788:
787:
780:
777:
776:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
747:
746:
736:
733:
730:
729:
727:
721:
718:
716:
713:
712:
709:
706:
704:
701:
699:
696:
695:
692:
686:
684:
678:
676:
670:
668:
662:
660:
654:
653:
649:
645:
642:
641:
637:
633:
630:
629:
625:
621:
618:
617:
613:
609:
606:
605:
601:
597:
594:
593:
589:
585:
582:
581:
577:
573:
570:
569:
565:
561:
558:
557:
554:
551:
549:
546:
545:
542:
538:
533:
532:
525:
522:
520:
517:
515:
512:
511:
483:
458:
457:
455:
449:
446:
421:
418:
417:
411:
408:
406:
403:
402:
398:
397:
386:
383:
381:
378:
375:
372:
371:
370:
369:
366:
363:
362:
357:
354:
353:
350:
347:
346:
343:
340:
338:
336:
332:
331:
328:
325:
323:
320:
319:
316:
313:
311:
308:
307:
306:
305:
299:
296:
293:
288:
285:
284:
280:
275:
272:
269:
264:
261:
258:
253:
250:
249:
248:
247:
242:
241:Finite groups
237:
236:
225:
222:
220:
217:
216:
215:
214:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
169:
166:
164:
161:
159:
156:
155:
154:
153:
148:
145:
143:
140:
139:
138:
137:
134:
133:
129:
128:
123:
120:
118:
115:
113:
110:
108:
105:
102:
100:
97:
96:
95:
94:
89:
86:
84:
81:
79:
76:
75:
74:
73:
68:Basic notions
65:
64:
60:
56:
55:
52:
47:
43:
39:
38:
33:
19:
4460:Group theory
4434:Applications
4361:Circle group
4245:
4241:
4232:
4228:
4161:Conway group
4123:Cyclic group
3998:
3943:
3924:
3895:
3864:
3845:
3823:
3801:
3782:Gallian 2013
3777:
3765:
3752:
3740:
3728:
3723:, p. 4.
3701:
3689:
3682:Gallian 2013
3677:
3616:vector space
3562:integer ring
3517:
3487:
3479:
3412:Cyclic group
3351:Cyclic group
2989:
2902:
2893:Cayley table
2881:permutations
2868:
2821:
2811:
2805:
2536:Cayley table
2529:
2438:cyclic group
2431:
2389:
2385:
2382:
2375:
2371:
2367:
2363:
2359:Right cosets
2358:
2357:
2343:) must be a
2306:
2220:
2140:
2133:
2112:
2108:
2101:
2097:
2085:
2081:
2077:
2073:
2066:
2052:
2022:
2015:
2008:
1894:, while the
1890:subgroup of
1881:
1868:generated by
1867:
1844:
1804:
1793:
1784:
1780:
1765:
1711:
1701:
1676:
1655:
1483:
1479:
1473:
1469:
1385:intersection
1378:homomorphism
1364:
1360:
1340:
1336:
1332:
1323:
1319:
1315:
1279:
1272:
1223:
1209:
1200:
1196:
1179:
1175:
1173:
1166:
1137:
1121:
1099:
1086:
1062:
1022:
1006:
997:
983:
976:
972:
958:
954:
948:
944:
923:
921:
916:
912:
910:
896:
892:
882:
878:
853:
852:is called a
829:
821:group theory
818:
647:
635:
623:
611:
599:
587:
575:
563:
334:
291:
278:
267:
256:
252:Cyclic group
130:
117:Free product
88:Group action
77:
51:Group theory
46:Group theory
45:
4090:Point group
4085:Space group
2986:12 elements
2899:24 elements
2887:Each group
2323:denote the
1693:of a group
962:, read as "
934:which is a
926:of a group
900:, read as "
874:restriction
841: ∗, a
825:mathematics
537:Topological
376:alternating
4454:Categories
4402:Loop group
4262:Lie groups
4034:direct sum
3791:References
3706:Artin 2011
3599:in a ring
3476:2 elements
3399:3 elements
3282:4 elements
3171:6 elements
3146:Subgroups:
3114:Subgroups:
3082:Subgroups:
3078:of order 8
3062:8 elements
3030:Subgroups:
2956:Simplified
2393:for every
1719:isomorphic
1407:-axis and
1313:such that
1194:, the sum
942:(that is,
644:Symplectic
584:Orthogonal
541:Lie groups
448:Free group
173:continuous
112:Direct sum
3915:807255720
3883:248917264
3514:1 element
3268:Subgroup:
3244:Subgroup:
3220:Subgroup:
3196:Subgroup:
2814:= {0, 4}
2169:−
2118:bijection
2105:given by
2096:φ :
2057:and some
1564:-axis in
1505:∪
1411:-axis in
1013:semigroup
998:overgroup
979:}
708:Conformal
596:Euclidean
203:nilpotent
18:Subgroups
3999:Subgroup
3893:(2013).
3844:(2011),
3822:(1974),
3800:(2009),
3745:Ash 2002
3638:See also
2826:, where
2370: :
2090:Because
2080: :
2005:itself,
1864:supremum
1807:⟩
1803:⟨
1714:⟩
1710:⟨
1704:⟩
1700:⟨
1679:⟩
1675:⟨
1658:⟩
1654:⟨
1242:identity
876:of ∗ to
854:subgroup
837:under a
703:Poincaré
548:Solenoid
420:Integers
410:Lattices
385:sporadic
380:Lie type
208:solvable
198:dihedral
183:additive
168:infinite
78:Subgroup
4429:History
3846:Algebra
3824:Algebra
3590:
3565:
3560:of the
3558:
3533:
2965:of the
2875:is the
2436:be the
2401:, then
2345:divisor
2196:
2151:
1902:itself.
1896:maximum
1888:minimum
1886:is the
1860:infimum
1837:
1812:
1774:, then
1758:
1723:
1672:is in
1631:
1599:
1595:
1566:
1554:
1529:
1525:
1489:
1442:
1413:
1330:, then
1291:inverse
1270:, then
992:, then
872:if the
698:Lorentz
620:Unitary
519:Lattice
459:PSL(2,
193:abelian
104:(Semi-)
4204:22..24
4156:22..24
4152:11..12
3983:Groups
3913:
3903:
3881:
3871:
3852:
3830:
3808:
3756:See a
3621:In an
3135:
3103:
2856:cyclic
2534:. Its
2353:|
2349:|
2325:orders
2321:|
2317:|
2313:|
2309:|
2307:where
2198:is in
2019:, and
1788:, and
1301:, and
1255:, and
1227:is in
1208:, and
1204:is in
1138:finite
1125:is in
1103:is in
1081:is in
1059:closed
843:subset
553:Circle
484:SL(2,
373:cyclic
337:-group
188:cyclic
163:finite
158:simple
142:kernel
4417:Sp(∞)
4414:SU(∞)
4308:Sp(n)
4302:SU(n)
4290:SO(n)
4278:SL(n)
4272:GL(n)
4025:Semi-
3670:Notes
3627:order
3614:of a
3597:ideal
2315:and
2208:index
2116:is a
2069:coset
2067:left
2045:Coset
1839:then
1800:. If
1794:order
1707:. If
1457:union
1132:When
1043:Then
957:<
832:group
737:Sp(∞)
734:SU(∞)
147:image
4411:O(∞)
4296:U(n)
4284:O(n)
4165:1..3
3911:OCLC
3901:ISBN
3879:OCLC
3869:ISBN
3850:ISBN
3828:ISBN
3806:ISBN
3518:The
3486:{1,
2990:The
2816:and
2432:Let
2331:and
2111:) =
2047:and
2021:3 +
2014:2 +
2007:1 +
1951:the
1764:mod
1463:and
1455:The
1391:and
1383:The
1305:and
1289:The
1240:The
1186:and
1111:and
1069:and
911:The
731:O(∞)
720:Loop
539:and
3930:doi
3595:An
3004:of
2969:of
2854:is
2538:is
2397:in
2383:If
2374:in
2366:= {
2347:of
2327:of
2232:,
2214:in
2210:of
2084:in
2076:= {
2061:in
1796:of
1721:to
1717:is
1635:If
1477:or
1395:of
1372:of
1350:If
1216:in
1190:in
1164:is
1148:of
1136:is
1115:in
1093:in
1085:.
1073:in
1000:of
984:If
981:).
975:≠ {
938:of
908:".
860:if
856:of
848:of
819:In
646:Sp(
634:SU(
610:SO(
574:SL(
562:GL(
4456::
4190:,
4183:,
4176:,
4163:Co
4154:,M
4027:)
3909:.
3877:.
3713:^
3610:A
3011:.
2895:.
2800:6
2776:7
2771:2
2747:3
2742:4
2718:5
2713:0
2689:1
2684:5
2672:4
2660:6
2655:1
2643:0
2631:2
2626:3
2614:2
2608:0
2602:4
2597:7
2585:6
2579:4
2573:0
2568:7
2565:3
2562:5
2559:1
2556:6
2553:2
2550:4
2547:0
2544:+
2390:Ha
2388:=
2386:aH
2378:}.
2368:ha
2364:Ha
2355:.
2139:~
2113:ah
2107:φ(
2102:aH
2100:→
2088:}.
2078:ah
2074:aH
2012:,
2001::
1884:}
1783:=
1482:⊆
1472:⊆
1363:→
1339:=
1337:ba
1335:=
1333:ab
1322:=
1320:ba
1318:=
1316:ab
1278:=
1231:.
1199:+
1122:ab
1079:ab
1004:.
947:≠
922:A
895:≤
881:×
622:U(
598:E(
586:O(
44:→
4353:8
4351:E
4345:7
4343:E
4337:6
4335:E
4329:4
4327:F
4321:2
4319:G
4246:n
4242:D
4233:n
4229:S
4217:M
4211:B
4202:F
4194:4
4192:J
4187:3
4185:J
4180:2
4178:J
4173:1
4171:J
4150:M
4136:n
4134:A
4127:n
4125:Z
4023:(
3975:e
3968:t
3961:v
3936:.
3932::
3917:.
3885:.
3859:.
3837:.
3815:.
3760:.
3747:.
3633:.
3607:.
3605:R
3601:R
3578::
3574:Z
3545:Z
3541:2
3488:p
3482:p
3465:3
3463:Z
3449:3
3447:Z
3433:3
3431:Z
3417:3
3415:Z
3388:4
3386:Z
3372:4
3370:Z
3356:4
3354:Z
3333:)
3329:(
3263:3
3261:S
3239:3
3237:S
3215:3
3213:S
3190:3
3188:S
3025:4
3023:A
3008:4
3006:S
2998:.
2973:4
2971:S
2925:4
2923:S
2907:4
2905:S
2872:4
2870:S
2864:4
2852:G
2848:H
2844:J
2840:G
2836:H
2832:H
2828:J
2822:H
2819:■
2812:J
2809:■
2797:2
2794:4
2791:0
2788:5
2785:1
2782:3
2779:7
2768:6
2765:0
2762:4
2759:1
2756:5
2753:7
2750:3
2739:0
2736:2
2733:6
2730:3
2727:7
2724:1
2721:5
2710:4
2707:6
2704:2
2701:7
2698:3
2695:5
2692:1
2681:1
2678:3
2675:7
2669:0
2666:2
2663:6
2652:5
2649:7
2646:3
2640:4
2637:6
2634:2
2623:7
2620:1
2617:5
2611:6
2605:4
2594:3
2591:5
2588:1
2582:2
2576:0
2514:}
2510:7
2507:,
2504:3
2501:,
2498:5
2495:,
2492:1
2489:,
2486:6
2483:,
2480:2
2477:,
2474:4
2471:,
2468:0
2464:{
2460:=
2457:G
2443:8
2441:Z
2434:G
2427:8
2419:p
2415:G
2411:p
2403:H
2399:G
2395:a
2376:H
2372:h
2351:G
2341:G
2337:G
2333:H
2329:G
2319:H
2311:G
2288:|
2284:H
2280:|
2273:|
2269:G
2265:|
2258:=
2255:]
2252:H
2249::
2246:G
2243:[
2230:H
2226:G
2216:G
2212:H
2204:H
2200:H
2182:2
2178:a
2172:1
2164:1
2160:a
2144:2
2141:a
2137:1
2134:a
2126:H
2122:G
2109:h
2098:H
2092:a
2086:H
2082:h
2063:G
2059:a
2055:H
2032:G
2023:H
2016:H
2009:H
2003:H
1999:H
1985:.
1981:Z
1977:2
1973:/
1968:Z
1957:H
1939:,
1935:Z
1931:8
1927:/
1922:Z
1911:G
1900:G
1892:G
1882:e
1880:{
1876:G
1872:e
1847:.
1841:a
1825:,
1821:Z
1805:a
1798:a
1790:n
1785:e
1781:a
1776:n
1772:n
1766:n
1760:(
1745:Z
1741:n
1737:/
1732:Z
1712:a
1702:a
1695:G
1691:a
1684:S
1677:S
1670:G
1665:S
1656:S
1649:S
1645:S
1641:G
1637:S
1619:.
1614:2
1609:R
1581:2
1576:R
1562:y
1558:x
1542:,
1538:Z
1512:Z
1508:3
1501:Z
1497:2
1484:A
1480:B
1474:B
1470:A
1465:B
1461:A
1452:.
1450:G
1446:G
1428:2
1423:R
1409:y
1405:x
1401:G
1397:G
1393:B
1389:A
1380:.
1374:H
1370:a
1365:G
1361:H
1356:G
1352:H
1347:.
1343:G
1341:e
1326:H
1324:e
1311:H
1307:b
1303:a
1299:G
1295:H
1286:.
1282:G
1280:e
1275:H
1273:e
1267:H
1265:e
1261:G
1257:H
1252:G
1250:e
1246:G
1229:H
1224:a
1222:−
1218:H
1214:a
1206:H
1201:b
1197:a
1192:H
1188:b
1184:a
1170:.
1167:a
1162:a
1158:n
1154:H
1150:H
1146:a
1142:H
1134:H
1127:H
1117:H
1113:b
1109:a
1105:H
1100:a
1095:H
1091:a
1083:H
1075:H
1071:b
1067:a
1055:H
1049:G
1045:H
1037:G
1033:G
1029:H
1025:G
1009:G
1002:H
994:G
990:G
986:H
977:e
973:H
968:G
964:H
959:G
955:H
949:G
945:H
940:G
932:H
928:G
917:e
906:G
902:H
897:G
893:H
888:H
883:H
879:H
870:G
866:H
862:H
858:G
850:G
846:H
835:G
808:e
801:t
794:v
690:8
688:E
682:7
680:E
674:6
672:E
666:4
664:F
658:2
656:G
650:)
648:n
638:)
636:n
626:)
624:n
614:)
612:n
602:)
600:n
590:)
588:n
578:)
576:n
566:)
564:n
506:)
493:Z
481:)
468:Z
444:)
431:Z
422:(
335:p
300:Q
292:n
289:D
279:n
276:A
268:n
265:S
257:n
254:Z
34:.
20:)
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