2968:
2100:
2963:{\displaystyle {\begin{array}{c|cccccc}&{\color {Blue}C_{id}}&{\color {Blue}C_{1}}&{\color {Blue}C_{g}}&{\color {Blue}C_{g^{2}}}&{\color {Gray}\dots }&{\color {Blue}C_{g^{p-2}}}\\\hline {\color {Blue}\chi _{1}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {2\pi i}{p-1}}}&{\color {Blue}e^{\frac {4\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {2\pi (p-2)i}{p-1}}}\\{\color {Blue}\chi _{2}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {4\pi i}{p-1}}}&{\color {Blue}e^{\frac {8\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {4\pi (p-2)i}{p-1}}}\\{\color {Blue}\chi _{3}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {6\pi i}{p-1}}}&{\color {Blue}e^{\frac {12\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {6\pi (p-2)i}{p-1}}}\\{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }\\{\color {Blue}\chi _{p-1}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}\dots }&{\color {Gray}1}\\{\color {Blue}\chi _{p}}&{\color {Gray}p-1}&{\color {Gray}-1}&{\color {Gray}0}&{\color {Gray}0}&{\color {Gray}\dots }&{\color {Gray}0}\end{array}}}
3573:
3048:
1707:
3568:{\displaystyle {\begin{aligned}{\text{1.}}&&(x,y)&\mapsto (x+a,y+b),\\{\text{2.}}&&(x,y)&\mapsto (ax,by),&\qquad {\text{where }}ab\neq 0,\\{\text{3.}}&&(x,y)&\mapsto (ax,y+b),&\qquad {\text{where }}a\neq 0,\\{\text{4.}}&&(x,y)&\mapsto (ax+y,ay),&\qquad {\text{where }}a\neq 0,\\{\text{5.}}&&(x,y)&\mapsto (x+y,y+a)\\{\text{6.}}&&(x,y)&\mapsto (a(x\cos t+y\sin t),a(-x\sin t+y\cos t)),&\qquad {\text{where }}a\neq 0.\end{aligned}}}
1365:
1702:{\displaystyle {\begin{aligned}C_{id}&=\left\{{\begin{pmatrix}1&0\\0&1\end{pmatrix}}\right\}\,,\\C_{1}&=\left\{{\begin{pmatrix}1&b\\0&1\end{pmatrix}}{\Bigg |}b\in \mathbf {F} _{p}^{*}\right\}\,,\\{\Bigg \{}C_{a}&=\left\{{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\Bigg |}b\in \mathbf {F} _{p}\right\}{\Bigg |}a\in \mathbf {F} _{p}\setminus \{0,1\}{\Bigg \}}\,.\end{aligned}}}
1337:
963:
1140:
1089:
1939:
433:
1810:
3053:
664:
271:
3972:
The subgroup of the special affine group consisting of those transformations whose linear part has determinant 1 is the group of orientation- and volume-preserving maps. Algebraically, this group is a semidirect product
859:
1332:{\displaystyle {\begin{pmatrix}c&d\\0&1\end{pmatrix}}{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\begin{pmatrix}c&d\\0&1\end{pmatrix}}^{-1}={\begin{pmatrix}a&(1-a)d+bc\\0&1\end{pmatrix}}\,,}
171:
4435:
1370:
3844:
4241:
3016:
351:(which is uniquely defined), however, no particular subgroup is a natural choice, since no point is special – this corresponds to the multiple choices of transverse subgroup, or splitting of the
3760:
591:
2062:
974:
2105:
1832:
3042:
are real numbers (the given conditions insure that transformations are invertible, but not for making the classes distinct; for example, the identity belongs to all the classes).
54:
Over any field, the affine group may be viewed as a matrix group in a natural way. If the associated field of scalars is the real or complex field, then the affine group is a
4201:
4177:
4143:
4077:
4279:
4009:
4324:
3959:
3924:
360:
4348:
4029:
2081:
is the dimension of the last irreducible representation. Finally using the orthogonality of irreducible representations, we can complete the character table of
1746:
616:
958:{\displaystyle A=\left({\begin{array}{cc}1&0\\0&0\end{array}}\right),\qquad B=\left({\begin{array}{cc}0&1\\0&0\end{array}}\right)\,,}
201:
3649:
The proof may be done by first remarking that if an affine transformation has no fixed point, then the matrix of the associated linear map has an
3792:
120:
4381:
4659:
4623:
4598:
4573:
752:, namely the stabilizer of this affine plane; the above matrix formulation is the (transpose of) the realization of this, with the
4212:
3632:
belong to cases 1, 3, and 5. The transformations that do not preserve the orientation of the plane belong to cases 2 (with
4534:
2981:
4350:
with the translations. Geometrically, it is the subgroup of the affine group generated by the orthogonal reflections.
3721:
625:
51:), the affine group consists of those functions from the space to itself such that the image of every line is a line.
4651:
3929:
1084:{\displaystyle e^{aA+bB}=\left({\begin{array}{cc}e^{a}&{\tfrac {b}{a}}(e^{a}-1)\\0&1\end{array}}\right)\,.}
512:
305:
3629:
1998:
1934:{\displaystyle \rho _{k}{\begin{pmatrix}a&b\\0&1\end{pmatrix}}=\exp \left({\frac {2ikj\pi }{p-1}}\right)}
4679:
3019:
4505:(the addition and origin), but not necessarily scalar multiplication, and these groups differ if working over
4466:
4115:
3579:
4684:
3622:
40:
4453:
4441:
4182:
4158:
4124:
4058:
3866:
3716:
3682:
352:
107:
4260:
3976:
4689:
4294:
4048:
442:
with a vector space, the subgroup that stabilizes the origin (of the vector space) is the original
36:
3884:
3654:
3586:
471:
312:
is isomorphic to the general linear group of the same dimension (so the stabilizer of a point in
189:
91:
4655:
4619:
4594:
4569:
4365:
4359:
832:
796:
780:
4543:
4327:
428:{\displaystyle 1\to V\to V\rtimes \operatorname {GL} (V)\to \operatorname {GL} (V)\to 1\,.}
4461:
4457:
3786:
3598:
3590:
44:
4333:
4014:
3869:. In terms of the semi-direct product, the special affine group consists of all pairs
3030:, an affine coordinate system exists on which it has one of the following forms, where
4497:. Note that this containment is in general proper, since by "automorphisms" one means
4291:
is a subgroup of the affine group. Algebraically, this group is a semidirect product
4673:
4644:
4561:
4369:
4032:
3615:
3608:
3594:
1805:{\displaystyle \rho _{k}:\operatorname {Aff} (\mathbf {F} _{p})\to \mathbb {C} ^{*}}
4639:
4547:
4044:
3604:
Case 3 corresponds to a scaling in one direction and a translation in another one.
3023:
659:{\displaystyle \left({\begin{array}{c|c}M&v\\\hline 0&1\end{array}}\right)}
608:
76:
32:
1008:
920:
874:
343:
All these subgroups are conjugate, where conjugation is given by translation from
188:
is the natural one (linear transformations are automorphisms), so this defines a
3854:
3027:
48:
20:
831:
matrices representing the affine group in one dimension. It is a two-parameter
266:{\displaystyle \operatorname {Aff} (n,K)=K^{n}\rtimes \operatorname {GL} (n,K)}
4108:
3650:
4051:, the affine group can be easily specified. For example, Günter Ewald wrote:
3789:
by a vector representation", and, as above, one has the short exact sequence
817:
Each of these two classes of matrices is closed under matrix multiplication.
835:
340:: recall that if one fixes a point, an affine space becomes a vector space.
55:
3853:
The subset of all invertible affine transformations that preserve a fixed
4283:
4152:
3965:
is a linear transformation of whose determinant has absolute value 1 and
166:{\displaystyle \operatorname {Aff} (V)=V\rtimes \operatorname {GL} (V)}
814:
identity matrix with the bottom row replaced by a row of all ones.
4568:. Vol. 1. Berlin Heidelberg: Springer-Verlag. Section 2.7.6.
4430:{\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {O} (1,3)}
4257:
is a
Euclidean space (over the field of real numbers), the group
3589:
that may differ in two different directions. When working with a
328:); formally, it is the general linear group of the vector space
1740:
one-dimensional representations, decided by the homomorphism
4266:
4501:
automorphisms, i.e., they preserve the group structure on
4532:
Poole, David G. (November 1995). "The
Stochastic Group".
795:
matrix in which the entries in each column sum to 1. The
458:
Representing the affine group as a semidirect product of
3861:. (The transformations themselves are sometimes called
838:, so with merely two generators (Lie algebra elements),
3839:{\displaystyle 1\to V\to V\rtimes _{\rho }G\to G\to 1.}
3022:. More precisely, given an affine transformation of an
766:) blocks corresponding to the direct sum decomposition
4456:– certain discrete subgroups of the affine group on a
4236:{\displaystyle {\mathfrak {A}}\subset {\mathfrak {P}}}
1851:
1580:
1472:
1402:
1270:
1222:
1185:
1149:
1024:
16:
Group of all affine transformations of an affine space
4384:
4336:
4297:
4263:
4215:
4185:
4161:
4127:
4061:
4017:
3979:
3932:
3887:
3795:
3724:
3685:, one can produce an affine group, sometimes denoted
3051:
2984:
2103:
2001:
1835:
1749:
1368:
1143:
977:
862:
619:
515:
438:
In the case that the affine group was constructed by
363:
204:
123:
3011:{\displaystyle \operatorname {Aff} (2,\mathbb {R} )}
802:
for passing from the above kind to this kind is the
3785:: one can say that the affine group obtained is "a
4643:
4429:
4342:
4318:
4273:
4235:
4195:
4171:
4137:
4071:
4023:
4003:
3953:
3918:
3838:
3754:
3567:
3010:
2962:
2056:
1933:
1804:
1701:
1331:
1083:
957:
658:
585:
427:
265:
165:
3755:{\displaystyle \rho :G\to \operatorname {GL} (V)}
1729:irreducible representations. By above paragraph (
1686:
1643:
1613:
1548:
1505:
3893:
86:acting by translations, and the affine group of
4031:with the translations. It is generated by the
3711:More generally and abstractly, given any group
47:(where the associated field of scalars is the
43:from the space into itself. In the case of a
3865:.) This group is the affine analogue of the
586:{\displaystyle (v,M)\cdot (w,N)=(v+Mw,MN)\,.}
8:
3655:Jordan normal form theorem for real matrices
3625:when the coordinate axes are perpendicular.
1681:
1669:
2057:{\displaystyle p(p-1)=p-1+\chi _{p}^{2}\,,}
820:The simplest paradigm may well be the case
1730:
300:Given the affine group of an affine space
82:obtained by "forgetting" the origin, with
4391:
4386:
4383:
4335:
4296:
4265:
4264:
4262:
4227:
4226:
4217:
4216:
4214:
4187:
4186:
4184:
4163:
4162:
4160:
4129:
4128:
4126:
4063:
4062:
4060:
4016:
3978:
3931:
3905:
3888:
3886:
3815:
3794:
3723:
3547:
3427:
3366:
3345:
3282:
3261:
3201:
3177:
3120:
3056:
3052:
3050:
3001:
3000:
2983:
2949:
2940:
2931:
2922:
2910:
2895:
2885:
2879:
2868:
2859:
2850:
2841:
2832:
2823:
2807:
2801:
2790:
2781:
2772:
2763:
2754:
2745:
2736:
2688:
2682:
2673:
2642:
2636:
2605:
2599:
2590:
2581:
2571:
2565:
2517:
2511:
2502:
2471:
2465:
2434:
2428:
2419:
2410:
2400:
2394:
2346:
2340:
2331:
2300:
2294:
2263:
2257:
2248:
2239:
2229:
2223:
2203:
2198:
2192:
2183:
2171:
2166:
2160:
2150:
2144:
2134:
2128:
2115:
2109:
2104:
2102:
2050:
2044:
2039:
2000:
1895:
1846:
1840:
1834:
1796:
1792:
1791:
1778:
1773:
1754:
1748:
1691:
1685:
1684:
1660:
1655:
1642:
1641:
1630:
1625:
1612:
1611:
1575:
1557:
1547:
1546:
1538:
1527:
1522:
1517:
1504:
1503:
1467:
1449:
1437:
1397:
1377:
1369:
1367:
1325:
1265:
1253:
1217:
1180:
1144:
1142:
1077:
1042:
1023:
1015:
1007:
982:
976:
951:
919:
873:
861:
720:is naturally isomorphic to a subgroup of
624:
618:
579:
514:
472:by construction of the semidirect product
421:
362:
233:
203:
122:
3018:can take a simple form on a well-chosen
4524:
4478:
3628:The affine transformations without any
1666:
3926:, that is, the affine transformations
3770:, one gets an associated affine group
292:is matrix multiplication of a vector.
67:Construction from general linear group
2950:
2941:
2932:
2923:
2911:
2896:
2880:
2869:
2860:
2851:
2842:
2833:
2824:
2802:
2791:
2782:
2773:
2764:
2755:
2746:
2737:
2683:
2674:
2637:
2600:
2591:
2582:
2566:
2512:
2503:
2466:
2429:
2420:
2411:
2395:
2341:
2332:
2295:
2258:
2249:
2240:
2224:
2193:
2184:
2161:
2145:
2129:
2110:
7:
4593:. Belmont: Wadsworth. Section 4.12.
4618:. Belmont: Wadsworth. p. 241.
4228:
4218:
4188:
4164:
4130:
4079:of all projective collineations of
4064:
90:can be described concretely as the
4440:This example is very important in
4406:
2974:Planar affine group over the reals
195:In terms of matrices, one writes:
14:
4085:is a group which we may call the
3969:is any fixed translation vector.
3661:Other affine groups and subgroups
3653:equal to one, and then using the
1981:. Then compare with the order of
506:, and multiplication is given by
276:where here the natural action of
71:Concretely, given a vector space
4387:
1774:
1656:
1626:
1518:
827:, that is, the upper triangular
62:Relation to general linear group
4196:{\displaystyle {\mathfrak {P}}}
4172:{\displaystyle {\mathfrak {P}}}
4138:{\displaystyle {\mathfrak {A}}}
4072:{\displaystyle {\mathfrak {P}}}
4011:of the special linear group of
3546:
3344:
3260:
3176:
908:
596:This can be represented as the
4548:10.1080/00029890.1995.12004664
4424:
4412:
4307:
4301:
4274:{\displaystyle {\mathcal {E}}}
4179:consisting of all elements of
4004:{\displaystyle SL(V)\ltimes V}
3992:
3986:
3936:
3906:
3902:
3896:
3889:
3830:
3824:
3805:
3799:
3749:
3743:
3734:
3538:
3535:
3502:
3493:
3463:
3457:
3454:
3447:
3435:
3420:
3396:
3393:
3386:
3374:
3336:
3312:
3309:
3302:
3290:
3252:
3231:
3228:
3221:
3209:
3168:
3150:
3147:
3140:
3128:
3110:
3086:
3083:
3076:
3064:
3005:
2991:
2709:
2697:
2538:
2526:
2367:
2355:
2017:
2005:
1787:
1784:
1769:
1290:
1278:
1054:
1035:
576:
552:
546:
534:
528:
516:
415:
412:
406:
397:
394:
388:
373:
367:
260:
248:
223:
211:
160:
154:
136:
130:
1:
4535:American Mathematical Monthly
4319:{\displaystyle O(V)\ltimes V}
4281:of distance-preserving maps (
4249:Isometries of Euclidean space
3954:{\displaystyle x\mapsto Mx+v}
3593:these directions need not be
736:embedded as the affine plane
4047:and the projective group of
1965:is a generator of the group
1731:§ Matrix representation
4368:is the affine group of the
3919:{\displaystyle |\det(M)|=1}
3601:need not be perpendicular.
705:row of zeros, and 1 is the
4706:
4652:Cambridge University Press
4357:
3618:combined with a dilation.
3611:combined with a dilation.
1359:conjugacy classes, namely
4616:Geometry: An Introduction
4591:Geometry: An Introduction
3857:up to sign is called the
498:is a linear transform in
474:, the elements are pairs
4642:(1985). "Section VI.1".
3614:Case 5 corresponds to a
3607:Case 4 corresponds to a
3020:affine coordinate system
4043:Presuming knowledge of
709:identity block matrix.
75:, it has an underlying
4614:Ewald, Günter (1971).
4589:Ewald, Günter (1971).
4431:
4344:
4320:
4275:
4253:When the affine space
4237:
4197:
4173:
4139:
4116:hyperplane at infinity
4073:
4025:
4005:
3955:
3920:
3840:
3756:
3621:Case 6 corresponds to
3585:Case 2 corresponds to
3578:Case 1 corresponds to
3569:
3012:
2964:
2058:
1935:
1806:
1703:
1333:
1085:
959:
783:representation is any
698:column vector, 0 is a
660:
587:
429:
267:
167:
41:affine transformations
4432:
4345:
4321:
4276:
4238:
4198:
4174:
4140:
4095:. If we proceed from
4074:
4026:
4006:
3956:
3921:
3841:
3757:
3570:
3013:
2965:
2059:
1936:
1807:
1704:
1334:
1086:
960:
661:
588:
454:Matrix representation
430:
296:Stabilizer of a point
268:
168:
4454:Affine Coxeter group
4382:
4334:
4295:
4261:
4213:
4183:
4159:
4125:
4101:to the affine space
4059:
4015:
3977:
3930:
3885:
3867:special linear group
3859:special affine group
3849:Special affine group
3793:
3722:
3683:general linear group
3049:
2982:
2101:
1999:
1833:
1747:
1366:
1141:
975:
860:
617:
513:
361:
353:short exact sequence
202:
121:
108:general linear group
29:general affine group
4646:Groups and Geometry
4049:projective geometry
4039:Projective subgroup
3670:Given any subgroup
2049:
1532:
1095:Character table of
4427:
4340:
4316:
4271:
4233:
4193:
4169:
4135:
4069:
4021:
4001:
3951:
3916:
3836:
3766:on a vector space
3752:
3565:
3563:
3008:
2960:
2958:
2954:
2945:
2936:
2927:
2918:
2906:
2891:
2873:
2864:
2855:
2846:
2837:
2828:
2819:
2795:
2786:
2777:
2768:
2759:
2750:
2741:
2730:
2678:
2669:
2632:
2595:
2586:
2577:
2559:
2507:
2498:
2461:
2424:
2415:
2406:
2388:
2336:
2327:
2290:
2253:
2244:
2235:
2217:
2188:
2179:
2156:
2140:
2124:
2054:
2035:
1931:
1876:
1802:
1712:Then we know that
1699:
1697:
1605:
1516:
1497:
1427:
1329:
1319:
1247:
1210:
1174:
1081:
1071:
1033:
955:
945:
899:
656:
650:
583:
425:
263:
190:semidirect product
163:
92:semidirect product
39:of all invertible
4343:{\displaystyle V}
4024:{\displaystyle V}
3693:, analogously as
3550:
3430:
3369:
3348:
3285:
3264:
3204:
3180:
3123:
3059:
2727:
2666:
2629:
2556:
2495:
2458:
2385:
2324:
2287:
1925:
1032:
320:is isomorphic to
4697:
4665:
4649:
4630:
4629:
4611:
4605:
4604:
4586:
4580:
4579:
4558:
4552:
4551:
4529:
4512:
4510:
4504:
4496:
4483:
4460:that preserve a
4436:
4434:
4433:
4428:
4402:
4401:
4390:
4374:
4349:
4347:
4346:
4341:
4328:orthogonal group
4325:
4323:
4322:
4317:
4290:
4280:
4278:
4277:
4272:
4270:
4269:
4256:
4242:
4240:
4239:
4234:
4232:
4231:
4222:
4221:
4206:
4202:
4200:
4199:
4194:
4192:
4191:
4178:
4176:
4175:
4170:
4168:
4167:
4150:
4144:
4142:
4141:
4136:
4134:
4133:
4118:, we obtain the
4113:
4106:
4100:
4094:
4087:projective group
4084:
4078:
4076:
4075:
4070:
4068:
4067:
4030:
4028:
4027:
4022:
4010:
4008:
4007:
4002:
3968:
3964:
3960:
3958:
3957:
3952:
3925:
3923:
3922:
3917:
3909:
3892:
3880:
3845:
3843:
3842:
3837:
3820:
3819:
3784:
3769:
3765:
3761:
3759:
3758:
3753:
3714:
3707:
3692:
3680:
3645:
3638:
3574:
3572:
3571:
3566:
3564:
3551:
3548:
3433:
3431:
3428:
3372:
3370:
3367:
3349:
3346:
3288:
3286:
3283:
3265:
3262:
3207:
3205:
3202:
3181:
3178:
3126:
3124:
3121:
3062:
3060:
3057:
3041:
3037:
3033:
3017:
3015:
3014:
3009:
3004:
2978:The elements of
2969:
2967:
2966:
2961:
2959:
2955:
2946:
2937:
2928:
2919:
2907:
2892:
2890:
2889:
2874:
2865:
2856:
2847:
2838:
2829:
2820:
2818:
2817:
2796:
2787:
2778:
2769:
2760:
2751:
2742:
2731:
2729:
2728:
2726:
2715:
2689:
2679:
2670:
2668:
2667:
2665:
2654:
2643:
2633:
2631:
2630:
2628:
2617:
2606:
2596:
2587:
2578:
2576:
2575:
2560:
2558:
2557:
2555:
2544:
2518:
2508:
2499:
2497:
2496:
2494:
2483:
2472:
2462:
2460:
2459:
2457:
2446:
2435:
2425:
2416:
2407:
2405:
2404:
2389:
2387:
2386:
2384:
2373:
2347:
2337:
2328:
2326:
2325:
2323:
2312:
2301:
2291:
2289:
2288:
2286:
2275:
2264:
2254:
2245:
2236:
2234:
2233:
2218:
2216:
2215:
2214:
2213:
2189:
2180:
2178:
2177:
2176:
2175:
2157:
2155:
2154:
2141:
2139:
2138:
2125:
2123:
2122:
2107:
2093:
2080:
2063:
2061:
2060:
2055:
2048:
2043:
1991:
1980:
1979:
1978:
1964:
1960:
1950:
1940:
1938:
1937:
1932:
1930:
1926:
1924:
1913:
1896:
1881:
1880:
1845:
1844:
1825:
1811:
1809:
1808:
1803:
1801:
1800:
1795:
1783:
1782:
1777:
1759:
1758:
1739:
1733:), there exist
1728:
1724:
1708:
1706:
1705:
1700:
1698:
1690:
1689:
1665:
1664:
1659:
1647:
1646:
1640:
1636:
1635:
1634:
1629:
1617:
1616:
1610:
1609:
1562:
1561:
1552:
1551:
1537:
1533:
1531:
1526:
1521:
1509:
1508:
1502:
1501:
1454:
1453:
1436:
1432:
1431:
1385:
1384:
1358:
1354:
1338:
1336:
1335:
1330:
1324:
1323:
1261:
1260:
1252:
1251:
1215:
1214:
1179:
1178:
1133:
1122:
1107:
1090:
1088:
1087:
1082:
1076:
1072:
1047:
1046:
1034:
1025:
1020:
1019:
999:
998:
964:
962:
961:
956:
950:
946:
904:
900:
852:
845:
841:
830:
826:
813:
801:
794:
775:
765:
761:
751:
735:
731:
719:
708:
704:
697:
690:
686:
682:
672:
665:
663:
662:
657:
655:
651:
607:
592:
590:
589:
584:
505:
497:
493:
489:
485:
469:
461:
449:
434:
432:
431:
426:
350:
346:
339:
327:
319:
311:
303:
291:
287:
272:
270:
269:
264:
238:
237:
187:
183:
172:
170:
169:
164:
113:
105:
97:
89:
85:
81:
74:
4705:
4704:
4700:
4699:
4698:
4696:
4695:
4694:
4680:Affine geometry
4670:
4669:
4668:
4662:
4638:
4634:
4633:
4626:
4613:
4612:
4608:
4601:
4588:
4587:
4583:
4576:
4560:
4559:
4555:
4531:
4530:
4526:
4521:
4516:
4515:
4506:
4502:
4486:
4484:
4480:
4475:
4458:Euclidean space
4450:
4385:
4380:
4379:
4372:
4362:
4356:
4332:
4331:
4293:
4292:
4288:
4259:
4258:
4254:
4251:
4211:
4210:
4204:
4181:
4180:
4157:
4156:
4146:
4123:
4122:
4111:
4107:by declaring a
4102:
4096:
4090:
4080:
4057:
4056:
4041:
4013:
4012:
3975:
3974:
3966:
3962:
3928:
3927:
3883:
3882:
3870:
3851:
3811:
3791:
3790:
3787:group extension
3780:
3771:
3767:
3763:
3720:
3719:
3712:
3694:
3686:
3671:
3668:
3663:
3640:
3633:
3599:coordinate axes
3591:Euclidean plane
3562:
3561:
3544:
3450:
3432:
3424:
3423:
3389:
3371:
3363:
3362:
3342:
3305:
3287:
3279:
3278:
3258:
3224:
3206:
3198:
3197:
3174:
3143:
3125:
3117:
3116:
3079:
3061:
3047:
3046:
3039:
3035:
3031:
2980:
2979:
2976:
2957:
2956:
2947:
2938:
2929:
2920:
2908:
2893:
2881:
2876:
2875:
2866:
2857:
2848:
2839:
2830:
2821:
2803:
2798:
2797:
2788:
2779:
2770:
2761:
2752:
2743:
2733:
2732:
2716:
2690:
2684:
2680:
2671:
2655:
2644:
2638:
2634:
2618:
2607:
2601:
2597:
2588:
2579:
2567:
2562:
2561:
2545:
2519:
2513:
2509:
2500:
2484:
2473:
2467:
2463:
2447:
2436:
2430:
2426:
2417:
2408:
2396:
2391:
2390:
2374:
2348:
2342:
2338:
2329:
2313:
2302:
2296:
2292:
2276:
2265:
2259:
2255:
2246:
2237:
2225:
2220:
2219:
2199:
2194:
2190:
2181:
2167:
2162:
2158:
2146:
2142:
2130:
2126:
2111:
2099:
2098:
2091:
2082:
2073:
2068:
1997:
1996:
1990:
1982:
1977:
1972:
1971:
1970:
1966:
1962:
1952:
1945:
1914:
1897:
1891:
1875:
1874:
1869:
1863:
1862:
1857:
1847:
1836:
1831:
1830:
1816:
1790:
1772:
1750:
1745:
1744:
1734:
1726:
1722:
1713:
1696:
1695:
1654:
1624:
1604:
1603:
1598:
1592:
1591:
1586:
1576:
1574:
1570:
1563:
1553:
1543:
1542:
1496:
1495:
1490:
1484:
1483:
1478:
1468:
1466:
1462:
1455:
1445:
1442:
1441:
1426:
1425:
1420:
1414:
1413:
1408:
1398:
1393:
1386:
1373:
1364:
1363:
1356:
1352:
1343:
1318:
1317:
1312:
1306:
1305:
1276:
1266:
1246:
1245:
1240:
1234:
1233:
1228:
1218:
1216:
1209:
1208:
1203:
1197:
1196:
1191:
1181:
1173:
1172:
1167:
1161:
1160:
1155:
1145:
1139:
1138:
1124:
1120:
1111:
1109:
1105:
1096:
1070:
1069:
1064:
1058:
1057:
1038:
1021:
1011:
1003:
978:
973:
972:
944:
943:
938:
932:
931:
926:
915:
898:
897:
892:
886:
885:
880:
869:
858:
857:
847:
843:
839:
828:
821:
803:
799:
784:
767:
763:
753:
737:
733:
721:
713:
706:
699:
692:
688:
684:
674:
670:
649:
648:
643:
637:
636:
631:
620:
615:
614:
597:
511:
510:
499:
495:
491:
490:is a vector in
487:
475:
463:
459:
456:
443:
359:
358:
348:
344:
329:
321:
313:
309:
301:
298:
289:
277:
229:
200:
199:
185:
177:
119:
118:
111:
99:
95:
87:
83:
79:
72:
69:
64:
45:Euclidean space
17:
12:
11:
5:
4703:
4701:
4693:
4692:
4687:
4682:
4672:
4671:
4667:
4666:
4660:
4635:
4632:
4631:
4624:
4606:
4599:
4581:
4574:
4553:
4542:(9): 798–801.
4523:
4522:
4520:
4517:
4514:
4513:
4477:
4476:
4474:
4471:
4470:
4469:
4464:
4449:
4446:
4438:
4437:
4426:
4423:
4420:
4417:
4414:
4411:
4408:
4405:
4400:
4397:
4394:
4389:
4366:Poincaré group
4360:Poincaré group
4358:Main article:
4355:
4354:Poincaré group
4352:
4339:
4315:
4312:
4309:
4306:
4303:
4300:
4268:
4250:
4247:
4246:
4245:
4244:
4243:
4230:
4225:
4220:
4190:
4166:
4132:
4066:
4040:
4037:
4033:shear mappings
4020:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3950:
3947:
3944:
3941:
3938:
3935:
3915:
3912:
3908:
3904:
3901:
3898:
3895:
3891:
3863:equiaffinities
3850:
3847:
3835:
3832:
3829:
3826:
3823:
3818:
3814:
3810:
3807:
3804:
3801:
3798:
3776:
3751:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3717:representation
3667:
3664:
3662:
3659:
3576:
3575:
3560:
3557:
3554:
3545:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3453:
3451:
3449:
3446:
3443:
3440:
3437:
3434:
3426:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3390:
3388:
3385:
3382:
3379:
3376:
3373:
3365:
3364:
3361:
3358:
3355:
3352:
3343:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3306:
3304:
3301:
3298:
3295:
3292:
3289:
3281:
3280:
3277:
3274:
3271:
3268:
3259:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3225:
3223:
3220:
3217:
3214:
3211:
3208:
3200:
3199:
3196:
3193:
3190:
3187:
3184:
3175:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3144:
3142:
3139:
3136:
3133:
3130:
3127:
3119:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3080:
3078:
3075:
3072:
3069:
3066:
3063:
3055:
3054:
3007:
3003:
2999:
2996:
2993:
2990:
2987:
2975:
2972:
2971:
2970:
2953:
2948:
2944:
2939:
2935:
2930:
2926:
2921:
2917:
2914:
2909:
2905:
2902:
2899:
2894:
2888:
2884:
2878:
2877:
2872:
2867:
2863:
2858:
2854:
2849:
2845:
2840:
2836:
2831:
2827:
2822:
2816:
2813:
2810:
2806:
2800:
2799:
2794:
2789:
2785:
2780:
2776:
2771:
2767:
2762:
2758:
2753:
2749:
2744:
2740:
2735:
2734:
2725:
2722:
2719:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2687:
2681:
2677:
2672:
2664:
2661:
2658:
2653:
2650:
2647:
2641:
2635:
2627:
2624:
2621:
2616:
2613:
2610:
2604:
2598:
2594:
2589:
2585:
2580:
2574:
2570:
2564:
2563:
2554:
2551:
2548:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2516:
2510:
2506:
2501:
2493:
2490:
2487:
2482:
2479:
2476:
2470:
2464:
2456:
2453:
2450:
2445:
2442:
2439:
2433:
2427:
2423:
2418:
2414:
2409:
2403:
2399:
2393:
2392:
2383:
2380:
2377:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2345:
2339:
2335:
2330:
2322:
2319:
2316:
2311:
2308:
2305:
2299:
2293:
2285:
2282:
2279:
2274:
2271:
2268:
2262:
2256:
2252:
2247:
2243:
2238:
2232:
2228:
2222:
2221:
2212:
2209:
2206:
2202:
2197:
2191:
2187:
2182:
2174:
2170:
2165:
2159:
2153:
2149:
2143:
2137:
2133:
2127:
2121:
2118:
2114:
2108:
2106:
2087:
2071:
2065:
2064:
2053:
2047:
2042:
2038:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
1986:
1973:
1942:
1941:
1929:
1923:
1920:
1917:
1912:
1909:
1906:
1903:
1900:
1894:
1890:
1887:
1884:
1879:
1873:
1870:
1868:
1865:
1864:
1861:
1858:
1856:
1853:
1852:
1850:
1843:
1839:
1813:
1812:
1799:
1794:
1789:
1786:
1781:
1776:
1771:
1768:
1765:
1762:
1757:
1753:
1718:
1710:
1709:
1694:
1688:
1683:
1680:
1677:
1674:
1671:
1668:
1663:
1658:
1653:
1650:
1645:
1639:
1633:
1628:
1623:
1620:
1615:
1608:
1602:
1599:
1597:
1594:
1593:
1590:
1587:
1585:
1582:
1581:
1579:
1573:
1569:
1566:
1564:
1560:
1556:
1550:
1545:
1544:
1541:
1536:
1530:
1525:
1520:
1515:
1512:
1507:
1500:
1494:
1491:
1489:
1486:
1485:
1482:
1479:
1477:
1474:
1473:
1471:
1465:
1461:
1458:
1456:
1452:
1448:
1444:
1443:
1440:
1435:
1430:
1424:
1421:
1419:
1416:
1415:
1412:
1409:
1407:
1404:
1403:
1401:
1396:
1392:
1389:
1387:
1383:
1380:
1376:
1372:
1371:
1348:
1340:
1339:
1328:
1322:
1316:
1313:
1311:
1308:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1275:
1272:
1271:
1269:
1264:
1259:
1256:
1250:
1244:
1241:
1239:
1236:
1235:
1232:
1229:
1227:
1224:
1223:
1221:
1213:
1207:
1204:
1202:
1199:
1198:
1195:
1192:
1190:
1187:
1186:
1184:
1177:
1171:
1168:
1166:
1163:
1162:
1159:
1156:
1154:
1151:
1150:
1148:
1116:
1108:
1101:
1093:
1092:
1091:
1080:
1075:
1068:
1065:
1063:
1060:
1059:
1056:
1053:
1050:
1045:
1041:
1037:
1031:
1028:
1022:
1018:
1014:
1010:
1009:
1006:
1002:
997:
994:
991:
988:
985:
981:
966:
965:
954:
949:
942:
939:
937:
934:
933:
930:
927:
925:
922:
921:
918:
914:
911:
907:
903:
896:
893:
891:
888:
887:
884:
881:
879:
876:
875:
872:
868:
865:
667:
666:
654:
647:
644:
642:
639:
638:
635:
632:
630:
627:
626:
623:
594:
593:
582:
578:
575:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
455:
452:
436:
435:
424:
420:
417:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
297:
294:
274:
273:
262:
259:
256:
253:
250:
247:
244:
241:
236:
232:
228:
225:
222:
219:
216:
213:
210:
207:
176:The action of
174:
173:
162:
159:
156:
153:
150:
147:
144:
141:
138:
135:
132:
129:
126:
68:
65:
63:
60:
15:
13:
10:
9:
6:
4:
3:
2:
4702:
4691:
4688:
4686:
4683:
4681:
4678:
4677:
4675:
4663:
4661:0-521-31694-4
4657:
4653:
4648:
4647:
4641:
4640:Lyndon, Roger
4637:
4636:
4627:
4625:9780534000349
4621:
4617:
4610:
4607:
4602:
4600:9780534000349
4596:
4592:
4585:
4582:
4577:
4575:9780534000349
4571:
4567:
4563:
4557:
4554:
4549:
4545:
4541:
4537:
4536:
4528:
4525:
4518:
4509:
4500:
4494:
4490:
4482:
4479:
4472:
4468:
4465:
4463:
4459:
4455:
4452:
4451:
4447:
4445:
4443:
4421:
4418:
4415:
4409:
4403:
4398:
4395:
4392:
4378:
4377:
4376:
4371:
4370:Lorentz group
4367:
4361:
4353:
4351:
4337:
4329:
4313:
4310:
4304:
4298:
4286:
4285:
4248:
4223:
4209:
4208:
4154:
4149:
4121:
4117:
4110:
4105:
4099:
4093:
4088:
4083:
4054:
4053:
4052:
4050:
4046:
4038:
4036:
4034:
4018:
3998:
3995:
3989:
3983:
3980:
3970:
3948:
3945:
3942:
3939:
3933:
3913:
3910:
3899:
3878:
3874:
3868:
3864:
3860:
3856:
3848:
3846:
3833:
3827:
3821:
3816:
3812:
3808:
3802:
3796:
3788:
3783:
3779:
3774:
3746:
3740:
3737:
3731:
3728:
3725:
3718:
3709:
3706:
3702:
3698:
3690:
3684:
3678:
3674:
3665:
3660:
3658:
3656:
3652:
3647:
3643:
3639:) or 3 (with
3636:
3631:
3626:
3624:
3619:
3617:
3616:shear mapping
3612:
3610:
3609:shear mapping
3605:
3602:
3600:
3596:
3595:perpendicular
3592:
3588:
3583:
3581:
3558:
3555:
3552:
3541:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3499:
3496:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3460:
3452:
3444:
3441:
3438:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3391:
3383:
3380:
3377:
3359:
3356:
3353:
3350:
3339:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3307:
3299:
3296:
3293:
3275:
3272:
3269:
3266:
3255:
3249:
3246:
3243:
3240:
3237:
3234:
3226:
3218:
3215:
3212:
3194:
3191:
3188:
3185:
3182:
3171:
3165:
3162:
3159:
3156:
3153:
3145:
3137:
3134:
3131:
3113:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3081:
3073:
3070:
3067:
3045:
3044:
3043:
3029:
3025:
3021:
2997:
2994:
2988:
2985:
2973:
2951:
2942:
2933:
2924:
2915:
2912:
2903:
2900:
2897:
2886:
2882:
2870:
2861:
2852:
2843:
2834:
2825:
2814:
2811:
2808:
2804:
2792:
2783:
2774:
2765:
2756:
2747:
2738:
2723:
2720:
2717:
2712:
2706:
2703:
2700:
2694:
2691:
2685:
2675:
2662:
2659:
2656:
2651:
2648:
2645:
2639:
2625:
2622:
2619:
2614:
2611:
2608:
2602:
2592:
2583:
2572:
2568:
2552:
2549:
2546:
2541:
2535:
2532:
2529:
2523:
2520:
2514:
2504:
2491:
2488:
2485:
2480:
2477:
2474:
2468:
2454:
2451:
2448:
2443:
2440:
2437:
2431:
2421:
2412:
2401:
2397:
2381:
2378:
2375:
2370:
2364:
2361:
2358:
2352:
2349:
2343:
2333:
2320:
2317:
2314:
2309:
2306:
2303:
2297:
2283:
2280:
2277:
2272:
2269:
2266:
2260:
2250:
2241:
2230:
2226:
2210:
2207:
2204:
2200:
2195:
2185:
2172:
2168:
2163:
2151:
2147:
2135:
2131:
2119:
2116:
2112:
2097:
2096:
2095:
2090:
2086:
2078:
2074:
2051:
2045:
2040:
2036:
2032:
2029:
2026:
2023:
2020:
2014:
2011:
2008:
2002:
1995:
1994:
1993:
1989:
1985:
1976:
1969:
1959:
1955:
1948:
1927:
1921:
1918:
1915:
1910:
1907:
1904:
1901:
1898:
1892:
1888:
1885:
1882:
1877:
1871:
1866:
1859:
1854:
1848:
1841:
1837:
1829:
1828:
1827:
1823:
1819:
1797:
1779:
1766:
1763:
1760:
1755:
1751:
1743:
1742:
1741:
1737:
1732:
1721:
1717:
1692:
1678:
1675:
1672:
1661:
1651:
1648:
1637:
1631:
1621:
1618:
1606:
1600:
1595:
1588:
1583:
1577:
1571:
1567:
1565:
1558:
1554:
1539:
1534:
1528:
1523:
1513:
1510:
1498:
1492:
1487:
1480:
1475:
1469:
1463:
1459:
1457:
1450:
1446:
1438:
1433:
1428:
1422:
1417:
1410:
1405:
1399:
1394:
1390:
1388:
1381:
1378:
1374:
1362:
1361:
1360:
1351:
1347:
1326:
1320:
1314:
1309:
1302:
1299:
1296:
1293:
1287:
1284:
1281:
1273:
1267:
1262:
1257:
1254:
1248:
1242:
1237:
1230:
1225:
1219:
1211:
1205:
1200:
1193:
1188:
1182:
1175:
1169:
1164:
1157:
1152:
1146:
1137:
1136:
1135:
1131:
1127:
1119:
1115:
1104:
1100:
1094:
1078:
1073:
1066:
1061:
1051:
1048:
1043:
1039:
1029:
1026:
1016:
1012:
1004:
1000:
995:
992:
989:
986:
983:
979:
971:
970:
969:
952:
947:
940:
935:
928:
923:
916:
912:
909:
905:
901:
894:
889:
882:
877:
870:
866:
863:
856:
855:
854:
851:
837:
834:
824:
818:
815:
811:
807:
798:
792:
788:
782:
777:
774:
770:
760:
756:
749:
745:
741:
729:
725:
717:
710:
703:
695:
681:
677:
652:
645:
640:
633:
628:
621:
613:
612:
611:
610:
605:
601:
580:
573:
570:
567:
564:
561:
558:
555:
549:
543:
540:
537:
531:
525:
522:
519:
509:
508:
507:
503:
483:
479:
473:
467:
453:
451:
447:
441:
422:
418:
409:
403:
400:
391:
385:
382:
379:
376:
370:
364:
357:
356:
355:
354:
341:
337:
333:
325:
317:
307:
295:
293:
285:
281:
257:
254:
251:
245:
242:
239:
234:
230:
226:
220:
217:
214:
208:
205:
198:
197:
196:
193:
191:
181:
157:
151:
148:
145:
142:
139:
133:
127:
124:
117:
116:
115:
109:
103:
93:
78:
66:
61:
59:
57:
52:
50:
46:
42:
38:
34:
30:
26:
22:
4685:Group theory
4645:
4615:
4609:
4590:
4584:
4565:
4556:
4539:
4533:
4527:
4507:
4498:
4492:
4488:
4481:
4439:
4363:
4282:
4252:
4147:
4120:affine group
4119:
4103:
4097:
4091:
4086:
4081:
4045:projectivity
4042:
3971:
3876:
3872:
3862:
3858:
3852:
3781:
3777:
3772:
3710:
3704:
3700:
3696:
3688:
3676:
3672:
3669:
3666:General case
3648:
3641:
3634:
3627:
3623:similarities
3620:
3613:
3606:
3603:
3597:, since the
3584:
3580:translations
3577:
3024:affine plane
2977:
2088:
2084:
2076:
2069:
2066:
1987:
1983:
1974:
1967:
1957:
1953:
1946:
1943:
1821:
1817:
1814:
1735:
1719:
1715:
1711:
1349:
1345:
1341:
1129:
1125:
1117:
1113:
1110:
1102:
1098:
967:
849:
846:, such that
822:
819:
816:
809:
805:
790:
786:
778:
772:
768:
758:
754:
747:
743:
739:
727:
723:
715:
711:
701:
693:
683:matrix over
679:
675:
668:
609:block matrix
603:
599:
595:
501:
481:
477:
465:
457:
445:
439:
437:
342:
335:
331:
323:
315:
299:
283:
279:
275:
194:
179:
175:
101:
77:affine space
70:
53:
49:real numbers
33:affine space
28:
25:affine group
24:
18:
4491:) < Aut(
4203:that leave
3855:volume form
3630:fixed point
3549:where
3347:where
3263:where
3179:where
833:non-Abelian
308:of a point
21:mathematics
4690:Lie groups
4674:Categories
4562:Berger, M.
4519:References
4442:relativity
4284:isometries
4109:hyperplane
3699:) :=
3651:eigenvalue
1992:, we have
1123:has order
797:similarity
712:Formally,
306:stabilizer
4467:Holomorph
4410:
4404:⋊
4311:⋉
4224:⊂
3996:⋉
3937:↦
3831:→
3825:→
3817:ρ
3813:⋊
3806:→
3800:→
3741:
3735:→
3726:ρ
3556:≠
3530:
3515:
3506:−
3488:
3473:
3455:↦
3394:↦
3354:≠
3310:↦
3270:≠
3229:↦
3189:≠
3148:↦
3084:↦
3026:over the
2989:
2943:…
2913:−
2901:−
2883:χ
2862:…
2812:−
2805:χ
2793:…
2784:…
2775:…
2766:…
2757:…
2748:…
2739:…
2721:−
2704:−
2695:π
2676:…
2660:−
2649:π
2623:−
2612:π
2569:χ
2550:−
2533:−
2524:π
2505:…
2489:−
2478:π
2452:−
2441:π
2398:χ
2379:−
2362:−
2353:π
2334:…
2318:−
2307:π
2281:−
2270:π
2227:χ
2208:−
2186:…
2037:χ
2027:−
2012:−
1919:−
1911:π
1889:
1838:ρ
1820:= 1, 2,…
1798:∗
1788:→
1767:
1752:ρ
1667:∖
1652:∈
1622:∈
1529:∗
1514:∈
1285:−
1255:−
1049:−
853:, where
836:Lie group
532:⋅
416:→
404:
398:→
386:
380:⋊
374:→
368:→
246:
240:⋊
209:
152:
146:⋊
128:
56:Lie group
4566:Geometry
4564:(1987).
4448:See also
4153:subgroup
4114:to be a
4055:The set
3675:< GL(
3587:scalings
1826:, where
1342:we know
1134:. Since
968:so that
808:+ 1) × (
789:+ 1) × (
602:+ 1) × (
486:, where
440:starting
4462:lattice
4326:of the
4207:fixed.
4151:as the
3681:of the
781:similar
742:, 1) |
732:, with
470:, then
314:Aff(2,
35:is the
31:of any
4658:
4622:
4597:
4572:
4485:Since
4373:O(1,3)
3961:where
3715:and a
3644:< 0
3637:< 0
3038:, and
2067:hence
673:is an
669:where
322:GL(2,
304:, the
106:, the
23:, the
4499:group
4473:Notes
4287:) of
3881:with
3028:reals
829:2 × 2
764:1 × 1
707:1 × 1
37:group
4656:ISBN
4620:ISBN
4595:ISBN
4570:ISBN
4364:The
3695:Aff(
3687:Aff(
2083:Aff(
1949:= −1
1944:and
1815:for
1725:has
1714:Aff(
1355:has
1344:Aff(
1132:− 1)
1112:Aff(
1097:Aff(
842:and
812:+ 1)
793:+ 1)
762:and
714:Aff(
700:1 ×
606:+ 1)
494:and
4544:doi
4540:102
4487:GL(
4330:of
4155:of
4145:of
4089:of
3894:det
3762:of
3646:).
3527:cos
3512:sin
3485:sin
3470:cos
2986:Aff
2079:− 1
1886:exp
1824:− 1
1764:Aff
1738:− 1
825:= 1
722:GL(
696:× 1
691:an
500:GL(
464:GL(
462:by
444:GL(
347:to
288:on
278:GL(
206:Aff
184:on
178:GL(
125:Aff
110:of
100:GL(
98:by
94:of
27:or
19:In
4676::
4654:.
4650:.
4538:.
4444:.
4375::
4035:.
3875:,
3834:1.
3738:GL
3708:.
3703:⋊
3657:.
3635:ab
3582:.
3559:0.
3429:6.
3368:5.
3284:4.
3203:3.
3122:2.
3058:1.
3034:,
2646:12
2094::
2075:=
1961:,
1956:=
1951:,
848:=
779:A
776:.
771:⊕
757:×
746:∈
738:{(
726:⊕
687:,
678:×
480:,
450:.
401:GL
383:GL
334:,
282:,
243:GL
192:.
149:GL
114::
58:.
4664:.
4628:.
4603:.
4578:.
4550:.
4546::
4511:.
4508:R
4503:V
4495:)
4493:V
4489:V
4425:)
4422:3
4419:,
4416:1
4413:(
4407:O
4399:3
4396:,
4393:1
4388:R
4338:V
4314:V
4308:)
4305:V
4302:(
4299:O
4289:A
4267:E
4255:A
4229:P
4219:A
4205:ω
4189:P
4165:P
4148:A
4131:A
4112:ω
4104:A
4098:P
4092:P
4082:P
4065:P
4019:V
3999:V
3993:)
3990:V
3987:(
3984:L
3981:S
3967:v
3963:M
3949:v
3946:+
3943:x
3940:M
3934:x
3914:1
3911:=
3907:|
3903:)
3900:M
3897:(
3890:|
3879:)
3877:v
3873:M
3871:(
3828:G
3822:G
3809:V
3803:V
3797:1
3782:G
3778:ρ
3775:⋊
3773:V
3768:V
3764:G
3750:)
3747:V
3744:(
3732:G
3729::
3713:G
3705:G
3701:V
3697:G
3691:)
3689:G
3679:)
3677:V
3673:G
3642:a
3553:a
3542:,
3539:)
3536:)
3533:t
3524:y
3521:+
3518:t
3509:x
3503:(
3500:a
3497:,
3494:)
3491:t
3482:y
3479:+
3476:t
3467:x
3464:(
3461:a
3458:(
3448:)
3445:y
3442:,
3439:x
3436:(
3421:)
3418:a
3415:+
3412:y
3409:,
3406:y
3403:+
3400:x
3397:(
3387:)
3384:y
3381:,
3378:x
3375:(
3360:,
3357:0
3351:a
3340:,
3337:)
3334:y
3331:a
3328:,
3325:y
3322:+
3319:x
3316:a
3313:(
3303:)
3300:y
3297:,
3294:x
3291:(
3276:,
3273:0
3267:a
3256:,
3253:)
3250:b
3247:+
3244:y
3241:,
3238:x
3235:a
3232:(
3222:)
3219:y
3216:,
3213:x
3210:(
3195:,
3192:0
3186:b
3183:a
3172:,
3169:)
3166:y
3163:b
3160:,
3157:x
3154:a
3151:(
3141:)
3138:y
3135:,
3132:x
3129:(
3114:,
3111:)
3108:b
3105:+
3102:y
3099:,
3096:a
3093:+
3090:x
3087:(
3077:)
3074:y
3071:,
3068:x
3065:(
3040:t
3036:b
3032:a
3006:)
3002:R
2998:,
2995:2
2992:(
2952:0
2934:0
2925:0
2916:1
2904:1
2898:p
2887:p
2871:1
2853:1
2844:1
2835:1
2826:1
2815:1
2809:p
2724:1
2718:p
2713:i
2710:)
2707:2
2701:p
2698:(
2692:6
2686:e
2663:1
2657:p
2652:i
2640:e
2626:1
2620:p
2615:i
2609:6
2603:e
2593:1
2584:1
2573:3
2553:1
2547:p
2542:i
2539:)
2536:2
2530:p
2527:(
2521:4
2515:e
2492:1
2486:p
2481:i
2475:8
2469:e
2455:1
2449:p
2444:i
2438:4
2432:e
2422:1
2413:1
2402:2
2382:1
2376:p
2371:i
2368:)
2365:2
2359:p
2356:(
2350:2
2344:e
2321:1
2315:p
2310:i
2304:4
2298:e
2284:1
2278:p
2273:i
2267:2
2261:e
2251:1
2242:1
2231:1
2211:2
2205:p
2201:g
2196:C
2173:2
2169:g
2164:C
2152:g
2148:C
2136:1
2132:C
2120:d
2117:i
2113:C
2092:)
2089:p
2085:F
2077:p
2072:p
2070:χ
2052:,
2046:2
2041:p
2033:+
2030:1
2024:p
2021:=
2018:)
2015:1
2009:p
2006:(
2003:p
1988:p
1984:F
1975:p
1968:F
1963:g
1958:g
1954:a
1947:i
1928:)
1922:1
1916:p
1908:j
1905:k
1902:i
1899:2
1893:(
1883:=
1878:)
1872:1
1867:0
1860:b
1855:a
1849:(
1842:k
1822:p
1818:k
1793:C
1785:)
1780:p
1775:F
1770:(
1761::
1756:k
1736:p
1727:p
1723:)
1720:p
1716:F
1693:.
1687:}
1682:}
1679:1
1676:,
1673:0
1670:{
1662:p
1657:F
1649:a
1644:|
1638:}
1632:p
1627:F
1619:b
1614:|
1607:)
1601:1
1596:0
1589:b
1584:a
1578:(
1572:{
1568:=
1559:a
1555:C
1549:{
1540:,
1535:}
1524:p
1519:F
1511:b
1506:|
1499:)
1493:1
1488:0
1481:b
1476:1
1470:(
1464:{
1460:=
1451:1
1447:C
1439:,
1434:}
1429:)
1423:1
1418:0
1411:0
1406:1
1400:(
1395:{
1391:=
1382:d
1379:i
1375:C
1357:p
1353:)
1350:p
1346:F
1327:,
1321:)
1315:1
1310:0
1303:c
1300:b
1297:+
1294:d
1291:)
1288:a
1282:1
1279:(
1274:a
1268:(
1263:=
1258:1
1249:)
1243:1
1238:0
1231:d
1226:c
1220:(
1212:)
1206:1
1201:0
1194:b
1189:a
1183:(
1176:)
1170:1
1165:0
1158:d
1153:c
1147:(
1130:p
1128:(
1126:p
1121:)
1118:p
1114:F
1106:)
1103:p
1099:F
1079:.
1074:)
1067:1
1062:0
1055:)
1052:1
1044:a
1040:e
1036:(
1030:a
1027:b
1017:a
1013:e
1005:(
1001:=
996:B
993:b
990:+
987:A
984:a
980:e
953:,
948:)
941:0
936:0
929:1
924:0
917:(
913:=
910:B
906:,
902:)
895:0
890:0
883:0
878:1
871:(
867:=
864:A
850:B
844:B
840:A
823:n
810:n
806:n
804:(
800:P
791:n
787:n
785:(
773:K
769:V
759:n
755:n
750:}
748:V
744:v
740:v
734:V
730:)
728:K
724:V
718:)
716:V
702:n
694:n
689:v
685:K
680:n
676:n
671:M
653:)
646:1
641:0
634:v
629:M
622:(
604:n
600:n
598:(
581:.
577:)
574:N
571:M
568:,
565:w
562:M
559:+
556:v
553:(
550:=
547:)
544:N
541:,
538:w
535:(
529:)
526:M
523:,
520:v
517:(
504:)
502:V
496:M
492:V
488:v
484:)
482:M
478:v
476:(
468:)
466:V
460:V
448:)
446:V
423:.
419:1
413:)
410:V
407:(
395:)
392:V
389:(
377:V
371:V
365:1
349:q
345:p
338:)
336:p
332:A
330:(
326:)
324:R
318:)
316:R
310:p
302:A
290:K
286:)
284:K
280:n
261:)
258:K
255:,
252:n
249:(
235:n
231:K
227:=
224:)
221:K
218:,
215:n
212:(
186:V
182:)
180:V
161:)
158:V
155:(
143:V
140:=
137:)
134:V
131:(
112:V
104:)
102:V
96:V
88:A
84:V
80:A
73:V
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