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群常识常用计算方法:

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发表于 2011-7-4 15:39 | 显示全部楼层 |阅读模式
[这个贴子最后由cjsh在 2011/07/04 03:48pm 第 2 次编辑] 因为老是似懂非懂,贴这当资料: 1 分解成不相交轮换乘积的过程 (1965)(1487)(1923)=(165)(234879) 1-----9-----9--------6 2-------3-------3-------3 3--------1--------4--------4 4--------4-----------8-----8 5---------5------5---------1 6--------6--------6------5 7--------7------1--------9 8-----------8------7--------7 9--------2----------2--------2 (165)(348792) 2 Z/群间求同态和环不同,环可用幂等元方便点,可群可用先求所有正规子群------KER,一个KER对应一种同态 HOM(Z4,Z6)={φ:x---->ax,a=0,3},x上带一横------剩余类 Z4------Z6 Lcm(4,6)/4=3;从3开始乘1,2,3。。。但最大不超6,所以就一个3*1 所以a=0,a=3 a=0是Z4 Z6------Z4 Lcm(4,6)/6=2;从2开始乘1,2,3。。。但最大不超4,所以就一个2*1 所以a=0,a=2 a=0是Z6 再找个大的 Z1050------------Z1500 Lcm(1050,1500)/1050=10,从10开始乘1,2,3。。。但最大不超1500,所以就10,2*10,3*10......149*10, 所以a=0,a=10.20,30....1480,1490 a=0是Z1050 共150个同态 Z1500------------Z1050 Lcm(1050,1500)/1500=7,从7开始乘1,2,3。。。但最大不超1050,所以就7,2*7,,3*7.。。。。。149*7, 所以a=0,a=7,14,21,。。。。。1043 a=0是Z1500 共150个同态 Z4--------Z6 Z6--------Z4 画横线的对应0x/3x 2x z:=IntegerRing(4) ; z4:=AdditiveGroup(z); z1:=IntegerRing(6) ; z6:=AdditiveGroup(z1); H44 := hom< z4 -> z4 | z4.1 -> z4.1 >; H44; H46:= hom< z4 -> z6 | z4.1 -> 0*z6.1 >;--------------------------------- H46; H46:= hom< z4 -> z6 | z4.1 -> 1*z6.1 >; H46; H46:= hom< z4 -> z6 | z4.1 -> 2*z6.1 >; H46; H46:= hom< z4 -> z6 | z4.1 -> 3*z6.1 >;--------------------------------- H46:= hom< z4 -> z6 | z4.1 -> 4*z6.1 >; H46:= hom< z4 -> z6 | z4.1 -> 5*z6.1 >; H46:= hom< z4 -> z6 | z4.1 -> 6*z6.1 >;--------------------------------- H46:= hom< z4 -> z6 | z4.1 ->7*z6.1 >; H46; H46:= hom< z4 -> z6 | z4.1 ->3*1999*z6.1 >;------------------------------ H46; H66:= hom< z6 -> z6 | z6.1 -> z6.1 >; H66; H64:= hom< z6 -> z4 | z6.1 -> 2*z4.1 >;-------------------------------- H64:= hom< z6 -> z4 | z6.1 -> 3*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 4*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 5*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 6*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 7*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 2*199*z4.1 >;------------------------------- Mapping from: GrpAb: z4 to GrpAb: z4 Mapping from: GrpAb: z4 to GrpAb: z6 >> H46:= hom< z4 -> z6 | z4.1 -> 1*z6.1 >; ^ Runtime error in hom< ... >: Images given do not define a homomorphism Mapping from: GrpAb: z4 to GrpAb: z6 >> H46:= hom< z4 -> z6 | z4.1 -> 2*z6.1 >; ^ Runtime error in hom< ... >: Images given do not define a homomorphism Mapping from: GrpAb: z4 to GrpAb: z6 >> H46:= hom< z4 -> z6 | z4.1 -> 4*z6.1 >; ^ Runtime error in hom< ... >: Images given do not define a homomorphism >> H46:= hom< z4 -> z6 | z4.1 -> 5*z6.1 >; ^ Runtime error in hom< ... >: Images given do not define a homomorphism >> H46:= hom< z4 -> z6 | z4.1 ->7*z6.1 >; ^ Runtime error in hom< ... >: Images given do not define a homomorphism Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z6 to GrpAb: z6 >> H64:= hom< z6 -> z4 | z6.1 -> 3*z4.1 >; ^ Runtime error in hom< ... >: Images given do not define a homomorphism Mapping from: GrpAb: z6 to GrpAb: z4 Mapping from: GrpAb: z6 to GrpAb: z4 >> H64:= hom< z6 -> z4 | z6.1 -> 5*z4.1 >; ^ Runtime error in hom< ... >: Images given do not define a homomorphism Mapping from: GrpAb: z6 to GrpAb: z4 Mapping from: GrpAb: z6 to GrpAb: z4 >> H64:= hom< z6 -> z4 | z6.1 -> 7*z4.1 >; ^ Runtime error in hom< ... >: Images given do not define a homomorphism Mapping from: GrpAb: z6 to GrpAb: z4 ============ 顺便连Z4--------Z*6 的也看了:都同态 Z*6 ----------z4 只有1x,3x,5x,不同态,2x,4x,6x都同态 Z*6 和Z6不同: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*(z*6).1 = 0 Abelian Group isomorphic to Z/6 Defined on 1 generator Relations: 6*z6.1 = 0 z:=IntegerRing(4) ; z4:=AdditiveGroup(z); z1:=IntegerRing(6) ; z6:=MultiplicativeGroup(z1); H44 := hom< z4 -> z4 | z4.1 -> z4.1 >; H44; H46:= hom< z4 -> z6 | z4.1 -> 0*z6.1 >; H46; H46:= hom< z4 -> z6 | z4.1 -> 1*z6.1 >; H46; H46:= hom< z4 -> z6 | z4.1 -> 2*z6.1 >; H46; H46:= hom< z4 -> z6 | z4.1 -> 3*z6.1 >; H46:= hom< z4 -> z6 | z4.1 -> 4*z6.1 >; H46:= hom< z4 -> z6 | z4.1 -> 5*z6.1 >; H46:= hom< z4 -> z6 | z4.1 -> 6*z6.1 >; H46:= hom< z4 -> z6 | z4.1 ->7*z6.1 >; H46; H46:= hom< z4 -> z6 | z4.1 ->3*1999*z6.1 >; H46; H66:= hom< z6 -> z6 | z6.1 -> z6.1 >; H66; H64:= hom< z6 -> z4 | z6.1 -> 2*z4.1 >; H64:= hom< z6 -> z4 | z6.1 -> 3*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 4*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 5*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 6*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 7*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 2*199*z4.1 >; Mapping from: GrpAb: z4 to GrpAb: z4 Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z6 to GrpAb: z6 >> H64:= hom< z6 -> z4 | z6.1 -> 3*z4.1 >; ^ Runtime error in hom< ... >: Images given do not define a homomorphism Mapping from: GrpAb: z6 to GrpAb: z4 Mapping from: GrpAb: z6 to GrpAb: z4 >> H64:= hom< z6 -> z4 | z6.1 -> 5*z4.1 >; ^ Runtime error in hom< ... >: Images given do not define a homomorphism Mapping from: GrpAb: z6 to GrpAb: z4 Mapping from: GrpAb: z6 to GrpAb: z4 >> H64:= hom< z6 -> z4 | z6.1 -> 7*z4.1 >; ^ Runtime error in hom< ... >: Images given do not define a homomorphism Mapping from: GrpAb: z6 to GrpAb: z4 ============== 顺便连Z*4--------Z*6 的也看了:都同态 有1x,3x,5x,,2x,4x,6x都同态 Z*6 ----------z4 有1x,3x,5x,,2x,4x,6x都同态 Z*6 和Z*4同构: NumberOfGenerators(z4) ; z1:=IntegerRing(6) ; z6:=MultiplicativeGroup(z1); z6; NumberOfGenerators(z4) ; IsIsomorphic(z4, z6); Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*z4.1 = 0 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*z6.1 = 0 1 true z:=IntegerRing(4) ; z4:=MultiplicativeGroup(z); z4; z1:=IntegerRing(6) ; z6:=MultiplicativeGroup(z1); z6; H44 := hom< z4 -> z4 | z4.1 -> z4.1 >; H44; H46:= hom< z4 -> z6 | z4.1 -> 0*z6.1 >; H46; H46:= hom< z4 -> z6 | z4.1 -> 1*z6.1 >; H46; H46:= hom< z4 -> z6 | z4.1 -> 2*z6.1 >; H46; H46:= hom< z4 -> z6 | z4.1 -> 3*z6.1 >; H46:= hom< z4 -> z6 | z4.1 -> 4*z6.1 >; H46:= hom< z4 -> z6 | z4.1 -> 5*z6.1 >; H46:= hom< z4 -> z6 | z4.1 -> 6*z6.1 >; H46:= hom< z4 -> z6 | z4.1 ->7*z6.1 >; H46; H46:= hom< z4 -> z6 | z4.1 ->3*1999*z6.1 >; H46; H66:= hom< z6 -> z6 | z6.1 -> z6.1 >; H66; H64:= hom< z6 -> z4 | z6.1 -> 2*z4.1 >; H64:= hom< z6 -> z4 | z6.1 -> 3*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 4*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 5*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 6*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 7*z4.1 >; H64; H64:= hom< z6 -> z4 | z6.1 -> 2*199*z4.1 >; Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*z4.1 = 0 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*z6.1 = 0 Mapping from: GrpAb: z4 to GrpAb: z4 Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z4 to GrpAb: z6 Mapping from: GrpAb: z6 to GrpAb: z6 Mapping from: GrpAb: z6 to GrpAb: z4 Mapping from: GrpAb: z6 to GrpAb: z4 Mapping from: GrpAb: z6 to GrpAb: z4 Mapping from: GrpAb: z6 to GrpAb: z4 Mapping from: GrpAb: z6 to GrpAb: z4 Z4:=AbelianGroup(GrpPerm,[4]); Z4; Z6:=AbelianGroup(GrpPerm,[6]); Z6; Z4Z4 := hom< Z4 -> Z4 | Z4.1 -> Z4.1 >; Z4Z4; Z4Z6 := hom< Z4 -> Z6 | Z4.1 -> Z6.1 >; Z4Z6; Z6Z4 := hom< Z6 -> Z4 | Z6.1 -> Z4.1 >; Z6Z4; Z6Z6 := hom< Z6 -> Z6 | Z6.1 -> Z6.1 >; Z6Z6; Z4Z4(Z4) eq Z4; Z6Z4(Z6) eq Z4; Z6Z4(Z6) eq Z4; Image(Z6Z6); Kernel(Z4Z4); Domain(Z4Z4); Domain(Z6Z6); Domain(Z6Z4); Domain(Z4Z6); Codomain(Z4Z4); Codomain(Z6Z6); Codomain(Z4Z6); Codomain(Z6Z4); DirectProduct(Z4, Z6) ; Order(1); Order(6); sub ; sub ; Index(Z4, Z6); Permutation group Z4 acting on a set of cardinality 4 Order = 4 = 2^2 (1, 2, 3, 4) Permutation group Z6 acting on a set of cardinality 6 Order = 6 = 2 * 3 (1, 2, 3, 4, 5, 6) Endomorphism of GrpPerm: Z4, Degree 4, Order 2^2 induced by (1, 2, 3, 4) |--> (1, 2, 3, 4) Homomorphism of GrpPerm: Z4, Degree 4, Order 2^2 into GrpPerm: Z6, Degree 6, Order 2 * 3 induced by (1, 2, 3, 4) |--> (1, 2, 3, 4, 5, 6) Homomorphism of GrpPerm: Z6, Degree 6, Order 2 * 3 into GrpPerm: Z4, Degree 4, Order 2^2 induced by (1, 2, 3, 4, 5, 6) |--> (1, 2, 3, 4) Endomorphism of GrpPerm: Z6, Degree 6, Order 2 * 3 induced by (1, 2, 3, 4, 5, 6) |--> (1, 2, 3, 4, 5, 6) true true true Permutation group acting on a set of cardinality 6 Order = 6 = 2 * 3 (1, 2, 3, 4, 5, 6) Permutation group acting on a set of cardinality 4 Order = 1 Permutation group Z4 acting on a set of cardinality 4 Order = 4 = 2^2 (1, 2, 3, 4) Permutation group Z6 acting on a set of cardinality 6 Order = 6 = 2 * 3 (1, 2, 3, 4, 5, 6) Permutation group Z6 acting on a set of cardinality 6 Order = 6 = 2 * 3 (1, 2, 3, 4, 5, 6) Permutation group Z4 acting on a set of cardinality 4 Order = 4 = 2^2 (1, 2, 3, 4) Permutation group Z4 acting on a set of cardinality 4 Order = 4 = 2^2 (1, 2, 3, 4) Permutation group Z6 acting on a set of cardinality 6 Order = 6 = 2 * 3 (1, 2, 3, 4, 5, 6) Permutation group Z6 acting on a set of cardinality 6 Order = 6 = 2 * 3 (1, 2, 3, 4, 5, 6) Permutation group Z4 acting on a set of cardinality 4 Order = 4 = 2^2 (1, 2, 3, 4) Permutation group acting on a set of cardinality 10 Order = 24 = 2^3 * 3 (1, 2, 3, 4) (5, 6, 7, 8, 9, 10) Integer Ring Integer Ring Permutation group acting on a set of cardinality 4 Id($) Mapping from: GrpPerm: $, Degree 4 to GrpPerm: Z4 [color=#0000FF]文字[color=#0000FF]文字文字
 楼主| 发表于 2011-7-4 15:44 | 显示全部楼层

群常识常用计算方法:

3 阿群初等因子不变因子: s1:=Sym({ 0..8 }); A0:= AbelianGroup([2^7,3^4,5,7]); A0; s2:=Sym({ 0..12 }); s2; A1:= AbelianGroup([2^10,3^5,5^2,7^11,13]); A1; A2:= AbelianGroup([30,140,250]); A2; Symmetric group s1 acting on a set of cardinality 9 Order = 362880 = 2^7 * 3^4 * 5 * 7 Abelian Group isomorphic to Z/362880 Defined on 4 generators Relations: 128*A0.1 = 0 81*A0.2 = 0 5*A0.3 = 0 7*A0.4 = 0 Symmetric group s2 acting on a set of cardinality 13 Order = 2^10 * 3^5 * 5^2 * 7 * 11 * 13 Abelian Group isomorphic to Z/159907204637107200 Defined on 5 generators Relations: 1024*A1.1 = 0 243*A1.2 = 0 25*A1.3 = 0 1977326743*A1.4 = 0 13*A1.5 = 0 Abelian Group isomorphic to Z/10 + Z/10 + Z/10500 Defined on 3 generators Relations: 30*A2.1 = 0 140*A2.2 = 0 250*A2.3 = 0 A2:= AbelianGroup([30,140,250]); A2; FactoredOrder(A2); Abelian Group isomorphic to Z/10 + Z/10 + Z/10500 Defined on 3 generators Relations: 30*A2.1 = 0 140*A2.2 = 0 250*A2.3 = 0 [ <2, 4>, <3, 1>, <5, 5>, <7, 1> ] 阶为1050000的群有:初等因子: 1:Z2+Z2+Z2+Z2+ Z3 +Z5+Z5+Z5+Z5+Z5 +Z7 初等群 2:Z4+Z2+Z2+ Z3 +Z5+Z5+Z5+Z5+Z5 +Z7 3:Z8+Z2+ Z3 +Z5+Z5+Z5+Z5+Z5 +Z7 4:Z16+ Z3 +Z5+Z5+Z5+Z5+Z5 +Z7 5:Z2+Z2+Z2+Z2+ Z3 +Z25+Z5+Z5+Z5 +Z7 6:Z2+Z2+Z2+Z2+ Z3 +Z75+Z5+Z5 +Z7 7:Z2+Z2+Z2+Z2+ Z3 +Z375+Z5 +Z7 8:Z2+Z2+Z2+Z2+ Z3 +Z375*5 +Z7 9:Z4+Z2+Z2+ Z3 +Z25+Z5+Z5+Z5 +Z7 ............. 共有4*5=20种 1:转不变因子: 12222 11113 55555 11117=Z5+Z/10+Z/10+Z/10+Z210 9:转不变因子: 1 2 2 4 1 1 1 3 5 5 5 25 1 1 1 7=Z5+Z/10+Z/10+Z/2100 验:对!同构可关系不同,1和9中一个是11/5生成元、一个11/4生成元 A22:= AbelianGroup([2,2,2,2,3,5,5,5,5,5,7]); A22; A22:= AbelianGroup([4,2,2,2,3,25,5,5,5,7]); A22; A2:= AbelianGroup([5,10,10,10,210]); A2; A22:= AbelianGroup([5,10,10,10,210]); A22; Abelian Group isomorphic to Z/5 + Z/10 + Z/10 + Z/10 + Z/210 Defined on 11 generators Relations: 2*A22.1 = 0 2*A22.2 = 0 2*A22.3 = 0 2*A22.4 = 0 3*A22.5 = 0 5*A22.6 = 0 5*A22.7 = 0 5*A22.8 = 0 5*A22.9 = 0 5*A22.10 = 0 7*A22.11 = 0 Abelian Group isomorphic to Z/10 + Z/10 + Z/10 + Z/2100 Defined on 10 generators Relations: 4*A22.1 = 0 2*A22.2 = 0 2*A22.3 = 0 2*A22.4 = 0 3*A22.5 = 0 25*A22.6 = 0 5*A22.7 = 0 5*A22.8 = 0 5*A22.9 = 0 7*A22.10 = 0 Abelian Group isomorphic to Z/5 + Z/10 + Z/10 + Z/10 + Z/210 Defined on 5 generators Relations: 5*A2.1 = 0 10*A2.2 = 0 10*A2.3 = 0 10*A2.4 = 0 210*A2.5 = 0 4 有限自由阿群及ECC中的RANK理解: 就是有整数无限群Z的,有一个Z,RANK=1,n个,RANK=n, 有限生成阿群就是有整数群Zm间直和,叫挠子群----都是阶有限的,在ECC里没MOD P前就是画直线只能求有限个新点, 有限生成阿群也有可能加几个无限群Z的,就叫有限生成自由阿群,象ECC里的曲线现在发现最高加了28个无限群Z,在ECC里就是画直线能求无限个新点,这28个点(生成元)如上 不过ECC曲线MOD P后就没无限群Z这直和项了 有限生成阿群麻烦在初等因子---就是Zm各项的m求法 也可表为不变因子之直和 初等因子就是素数次幂(》=1)分解之和----------不变因子转换麻烦点等看懂了贴这 RANK=3 E3:= EllipticCurve([0,0,0,-82,0]); E3; PointsAtInfinity(E3); TorsionSubgroup(E3); NumberOfGenerators(E3); Generators(E3) ; Rank(E3); MordellWeilShaInformation(E3); AbelianGroup(E3); P1:=E3![-8,-12]; Order(P1); P2:=E3![-1,-9]; Order(P2); P3:=E3![-9,-3]; Order(P3); P4:=E3![0,0]; Order(P4); P5:=E3![49/4,231/8]; Order(P5); P6:=E3![41/4,123/8]; Order(P6); Elliptic Curve defined by y^2 = x^3 - 82*x over Rational Field {@ (0 : 1 : 0) @} Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*$.1 = 0 4 [ (0 : 0 : 1), (-8 : 12 : 1), (-1 : -9 : 1), (-9 : -3 : 1) ] 3 Torsion Subgroup = Z/2 Analytic rank = 3 The 2-Selmer group has rank 4 Found a point of infinite order. Found 2 independent points. Found 3 independent points. After 2-descent: 3 <= Rank(E) <= 3 Sha(E)[2] is trivial (Searched up to height 100 on the 2-coverings.) [ 3, 3 ] [ (0 : 0 : 1), (49/4 : 231/8 : 1), (41/4 : 123/8 : 1), (-9 : 3 : 1) ] [ <2, [ 0, 0 ]> ] Abelian Group isomorphic to Z/2 + Z + Z + ZDefined on 4 generators Relations: 2*$.1 = 0 0 0就是无限远点,3个Z群,生成元P1/P2/P3 0 0 2 2阶点两个挠子群,生成元P4 0 还给了两能生成 Z群的点-----分数也行,只要是有理点 0 ----------- 100000*P4;挠子群Z转不出去 100001*P4; (0 : 1 : 0) (0 : 0 : 1) Z群就不同了,10倍就天文数字了,可至无穷啊 (5329/144 : 377191/1728 : 1) (-118764872/42003361 : -3937849795044/272223782641 : 1) (905925300579649/81949277077056 : 15640109412983368378849/741852714479323912704 : 1) (-2343465084196597805000/58051145063946951161569 : -25447354793050005176981463556943100/13986729827640029730757694665036753 : 1) (6302474098508073788197910531446609/651331880237428048545105511066896 : -176342732523930673371934739457017411658357216614489/16622774127029891762931364\ 252673114873466120250944 : 1) (-3813430085070199683943976557798873904463613832/228336826875138911896596598141\ 8615777603848641 : 125494893852863155426100974766561536605189928217522571232033\ 4800833324/10910984586392115360355313355505782235146810851274627664315031114256\ 1 : 1) (1880701588819492767318652030277859714069703191156088494052609/8018344141367449\ 6508774280969881993157804843783416953223424 : -237919946592218328913913464119432611871936626120863837945348807020008535118330\ 0600626847871/22705289196485300714253305436925878920371641314520424153731523057\ 840376182235816117587968 : 1) (-18815230184579826991959942950507242936389088171715617440267931518219770954888\ /2785944412729486384827083679528629259002420636194629323893461343681407696961 : 2305204200717457550192851363078188878716253530093747095851230611288478913625408\ 572457649696650868753188600387657492/147047847011317425399566847867848018869257\ 649673254707518241304784767107976145675764728694749353143581342134675041 : 1) (190909762816982266589447623951781585733423203877205763763861577911709314049308\ 99671515742264401/3759205612578986117009614770431984774241158894350139721932557\ 4936713501758087453032010090000 : -26373827153651235352076330133554362293020016\ 8338935545161185268632853079379474560027219993229788750399658301115811130372933\ 8142837271804656601/23048577162798624548776580990172190660088839559951272602553\ 3782892120933828081749097515442095457075887927302595340997837489878439223000000 : 1) 1000*P1;过5万位了 The output is too long and has been truncated. RANK=4 E3:= EllipticCurve([0,-1,0,- 24649,1355209]); E3; TorsionSubgroup(E3); NumberOfGenerators(E3); Generators(E3) ; NumberOfGenerators(E3); Rank(E3); AbelianGroup(E3); Elliptic Curve defined by y^2 = x^3 - x^2 - 24649*x + 1355209 over Rational Field Abelian Group isomorphic to Z/2 + Z/2 Defined on 2 generators Relations: 2*$.1 = 0 2*$.2 = 0 6 [ (67 : 0 : 1), (113 : 0 : 1), (149 : -984 : 1), (-15 : 1312 : 1), (313 : 4920 : 1), (-56 : -1599 : 1) ] 6 4 Abelian Group isomorphic to Z/2 + Z/2 + Z + Z + Z + ZDefined on 6 generators Relations: 2*$.1 = 0 2*$.2 = 0 E3:= EllipticCurve([0,-1,0,- 24649,1355209]); E3; Generators(E3) ; NumberOfGenerators(E3); Rank(E3); AbelianGroup(E3); Order(E3); P1:=E3![67,0]; Order(P1); P2:=E3![113,0]; Order(P2); P3:=E3![149,-984]; Order(P3); P4:=E3![-15,1312]; Order(P4); P5:=E3![313,4920]; Order(P5); P6:=E3![-56,-1599]; Order(P6); Elliptic Curve defined by y^2 = x^3 - x^2 - 24649*x + 1355209 over Rational Field [ (67 : 0 : 1), (113 : 0 : 1), (149 : -984 : 1), (-15 : 1312 : 1), (313 : 4920 : 1), (-56 : -1599 : 1) ] 6 4 Abelian Group isomorphic to Z/2 + Z/2 + Z + Z + Z + Z Defined on 6 generators Relations: 2*$.1 = 0 2*$.2 = 0 >> Order(E3); ^ Runtime error in ';Order';: Algorithm does not work for this ring 2------2阶点两个挠子群,生成元P1/P2 2 0 0 0 0-----------0就是无限远点,四个Z群,生成元P3/P4/P5/P6 [br][br]-=-=-=-=- 以下内容由 cjsh 时添加 -=-=-=-=- 一般线性群或者它的一些子群是非交换的,只有有限生成交换群采有秩。非交换的群是没有的。
 楼主| 发表于 2011-7-4 15:53 | 显示全部楼层

群常识常用计算方法:

[这个贴子最后由cjsh在 2011/07/04 04:09pm 第 1 次编辑] 5 非单非满同态是:A-------B非单非满同态时,A群的一个商群和B群的一个子群一样 φ(a+b)=φ(a)*φ(b)是对任意的a、b都必须成立,不是你说的a与b不能相同,它们是可以相同的,而且相同时也必须要成立才是同态。 单同态或满同态,是在首先确定是同态的基础上,再来判断映射的单或满。映射的单或满与群的结构毫无关系,判断单或满根本不用考虑群的运算。 你上边不应该把保持运算和单或满一起考虑,它们是独立的 你说的教科书上在同态前边加上非单非满四个字,这只是一个强调。一般说同态默认它是既不是单同态也不是满同态,所以你看到的外文书上没有这个词。它只是默认了。 6 稳定子群正好就是x的中心化子,也叫迷向子群 圈积!?! 中心是对群说的,中心化子是对群元素说的 7 2次域实例: K := QuadraticField(-5); K; K1 := QuadraticField(-51); K1; K := QuadraticField(-26); K; K := QuadraticField(-1136); K; K := QuadraticField(-50); K; C := ComplexField(5); Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field Quadratic Field with defining polynomial $.1^2 + 51 over the Rational Field Quadratic Field with defining polynomial $.1^2 + 26 over the Rational Field Quadratic Field with defining polynomial $.1^2 + 71 over the Rational Field Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field 12.056 - 4.7278*i 71 4.00000000000000000000000000000 8 格论: http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/ http://akpublic.research.att.com/~njas/lattices/index.html http://akpublic.research.att.com/~njas/lattices/index.html#An http://bbs.pediy.com/showthread.php?t=131611&page=6 印度人的群站 http://groupprops.subwiki.org/wiki/Main_Page http://groupprops.subwiki.org/wiki/Symmetric_group:S3 群论都提下俩本书,又有例子又全面 Contemporary Abstract Aralgeb, 2010 7th edition A_First_Course_in_Abstract_Algebra_7E_-_Fraleigh
 楼主| 发表于 2011-7-5 11:12 | 显示全部楼层

群常识常用计算方法:

9
单位群U(n)是循环群充要条件:
N=2,4,p,2p^m

循环群有本原根条件同
-----------
两循环群直和后两原生成元间叫桥
10
Z*(n)群是z(n)群的单位群
------------
循环群子群个数为n的不同正因子个数σ(n)
11
Z(n)-----------Z(m)求环同态:
先LCD(m,n)=k,然后求kx=0

[br][br]-=-=-=-=- 以下内容由 cjsh 时添加 -=-=-=-=-
S4 := Sym({ "a", "b", "c", "d"});
> S4;
f := NumberingMap(S4);
f;
[ [ f(x*y) : y in S4 ] : x in S4 ];
[br][br]-=-=-=-=- 以下内容由 cjsh 时添加 -=-=-=-=-
求凯莱表
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