JORDAN-HOLDER定理实例验证：

cjsh

S5------120阶不S4------24阶正规列长，先S5后S4，也贴些性质， S4一般书上都说了，把CALEY表也贴上，S5表太大了。。。 S5 := Sym({ "a", "b", "c", "d","F"});抽象群选了字母 > S5; Identity(S5); Degree(S5); Generators(S5);------------------两生成元 GeneratorsSequence(S5); NumberOfGenerators(S5); GSet(S5) ; Order(S5) ;-----------------群阶 FactoredOrder(S5);---------群阶分解 IsEven(S5);---------------偶置换吗？ IsSimple(S5) ;------------单群吗？ IsSoluble(S5);------------可解群吗？ Exponent(S5) ; NumberingMap(S5) ; Representative(S5) ; NumberOfClasses(S5) ; Centre(S5);---------------中心 SubgroupClasses(S5);------子群列19个 NormalSubgroups(S5);-------正规子群列3个 SolvableSubgroups(S5);------可解子群17个 SimpleSubgroups(S5);--------单子群就一个60阶的 DerivedSeries(S5);--------导群列 DerivedLength(S5) ;------导群列长 NormalLattice(S5) ; ChiefFactors(S5);-----------主列因子 ChiefSeries(S5);-----------主列 list := SmallGroups(120:Search:="Insoluble");列出不可解的就那一个60阶的 > #list; FittingSubgroup(S5) ; FrattiniSubgroup(S5); Radical(S5) ;------------诣零根，全部幂零元组成的理想，和判断群交环性有关。公式为RAD(R/RAD R)={0},或RAD(R/I)=(I^(1/2))/I 幂零元x作用：对交换环e-x是一个可逆元 RadicalQuotient(S5); CharacterTable(S5); PermutationCharacter(S5) ; AutomorphismGroup(S5); NameSimple(S5) ; Base(S5) ; Symmetric group S5 acting on a set of cardinality 5 Order = 120 = 2^3 * 3 * 5 Id(S5) 5 { (c, b), (c, b, a, d, F) } [ (c, b, a, d, F), (c, b) ] 2 GSet{@ c, b, a, d, F @} 120 [ <2, 3>, <3, 1>, <5, 1> ] false false false 60 Mapping from: GrpPerm: S5 to { 1 .. 120 } Id(S5) 7 Permutation group acting on a set of cardinality 5 Order = 1 Conjugacy classes of subgroups ------------------------------ [ 1] Order 1 Length 1 Permutation group acting on a set of cardinality 5 Order = 1 Id($) [ 2] Order 2 Length 10 Permutation group acting on a set of cardinality 5 Order = 2 (a, F) [ 3] Order 2 Length 15 Permutation group acting on a set of cardinality 5 Order = 2 (c, b)(a, F) [ 4] Order 3 Length 10 Permutation group acting on a set of cardinality 5 Order = 3 (c, a, F) [ 5] Order 5 Length 6 Permutation group acting on a set of cardinality 5 Order = 5 (c, a, b, F, d) [ 6] Order 4 Length 5 Permutation group acting on a set of cardinality 5 Order = 4 = 2^2 (c, b)(a, F) (c, a)(b, F) [ 7] Order 4 Length 15 Permutation group acting on a set of cardinality 5 Order = 4 = 2^2 (b, F, d, a) (b, d)(a, F) [ 8] Order 4 Length 15 Permutation group acting on a set of cardinality 5 Order = 4 = 2^2 (a, F) (b, d)(a, F) [ 9] Order 6 Length 10 Permutation group acting on a set of cardinality 5 Order = 6 = 2 * 3 (c, a)(d, F) (c, b, a) [10] Order 6 Length 10 Permutation group acting on a set of cardinality 5 Order = 6 = 2 * 3 (c, d, b) (a, F) [11] Order 6 Length 10 Permutation group acting on a set of cardinality 5 Order = 6 = 2 * 3 (a, F) (a, F, d) [12] Order 10 Length 6 Permutation group acting on a set of cardinality 5 Order = 10 = 2 * 5 (c, d)(a, F) (c, a, b, F, d) [13] Order 8 Length 15 Permutation group acting on a set of cardinality 5 Order = 8 = 2^3 (a, F) (b, d)(a, F) (b, F)(a, d) [14] Order 12 Length 5 Permutation group acting on a set of cardinality 5 Order = 12 = 2^2 * 3 (c, a, F) (c, b)(a, F) (c, a)(b, F) [15] Order 12 Length 10 Permutation group acting on a set of cardinality 5 Order = 12 = 2^2 * 3 (c, d) (c, d, b) (a, F) [16] Order 20 Length 6 Permutation group acting on a set of cardinality 5 Order = 20 = 2^2 * 5 (b, F, a, d) (b, a)(d, F) (c, d, a, b, F) [17] Order 24 Length 5 Permutation group acting on a set of cardinality 5 Order = 24 = 2^3 * 3 (a, F) (a, F, d) (b, d)(a, F) (b, F)(a, d) [18] Order 60 Length 1 Permutation group acting on a set of cardinality 5 Order = 60 = 2^2 * 3 * 5 (c, b)(d, F) (c, a, d) [19] Order 120 Length 1 Permutation group acting on a set of cardinality 5 Order = 120 = 2^3 * 3 * 5 (c, b) (c, d, F)(b, a) Conjugacy classes of subgroups ------------------------------ [1] Order 1 Length 1 Permutation group acting on a set of cardinality 5 Order = 1 Id($) [2] Order 60 Length 1 Permutation group acting on a set of cardinality 5 Order = 60 = 2^2 * 3 * 5 (c, b)(d, F) (c, a, d) [3] Order 120 Length 1 Permutation group acting on a set of cardinality 5 Order = 120 = 2^3 * 3 * 5 (c, b) (c, d, F)(b, a) Conjugacy classes of subgroups ------------------------------ [ 1] Order 1 Length 1 Permutation group acting on a set of cardinality 5 Order = 1 Id($) [ 2] Order 2 Length 10 Permutation group acting on a set of cardinality 5 Order = 2 (a, F) [ 3] Order 2 Length 15 Permutation group acting on a set of cardinality 5 Order = 2 (c, b)(a, F) [ 4] Order 3 Length 10 Permutation group acting on a set of cardinality 5 Order = 3 (c, a, F) [ 5] Order 5 Length 6 Permutation group acting on a set of cardinality 5 Order = 5 (c, a, b, F, d) [ 6] Order 4 Length 5 Permutation group acting on a set of cardinality 5 Order = 4 = 2^2 (c, b)(a, F) (c, a)(b, F) [ 7] Order 4 Length 15 Permutation group acting on a set of cardinality 5 Order = 4 = 2^2 (b, F, d, a) (b, d)(a, F) [ 8] Order 4 Length 15 Permutation group acting on a set of cardinality 5 Order = 4 = 2^2 (a, F) (b, d)(a, F) [ 9] Order 6 Length 10 Permutation group acting on a set of cardinality 5 Order = 6 = 2 * 3 (c, a)(d, F) (c, b, a) [10] Order 6 Length 10 Permutation group acting on a set of cardinality 5 Order = 6 = 2 * 3 (c, d, b) (a, F) [11] Order 6 Length 10 Permutation group acting on a set of cardinality 5 Order = 6 = 2 * 3 (a, F) (a, F, d) [12] Order 10 Length 6 Permutation group acting on a set of cardinality 5 Order = 10 = 2 * 5 (c, d)(a, F) (c, a, b, F, d) [13] Order 8 Length 15 Permutation group acting on a set of cardinality 5 Order = 8 = 2^3 (a, F) (b, d)(a, F) (b, F)(a, d) [14] Order 12 Length 5 Permutation group acting on a set of cardinality 5 Order = 12 = 2^2 * 3 (c, a, F) (c, b)(a, F) (c, a)(b, F) [15] Order 12 Length 10 Permutation group acting on a set of cardinality 5 Order = 12 = 2^2 * 3 (c, d) (c, d, b) (a, F) [16] Order 20 Length 6 Permutation group acting on a set of cardinality 5 Order = 20 = 2^2 * 5 (b, F, a, d) (b, a)(d, F) (c, d, a, b, F) [17] Order 24 Length 5 Permutation group acting on a set of cardinality 5 Order = 24 = 2^3 * 3 (a, F) (a, F, d) (b, d)(a, F) (b, F)(a, d) Conjugacy classes of subgroups ------------------------------ [1] Order 60 Length 1 Permutation group acting on a set of cardinality 5 Order = 60 = 2^2 * 3 * 5 (c, b)(d, F) (c, a, d) [ Symmetric group S5 acting on a set of cardinality 5 Order = 120 = 2^3 * 3 * 5 (c, b, a, d, F) (c, b), Permutation group acting on a set of cardinality 5 Order = 60 = 2^2 * 3 * 5 (c, b, a) (b, a, d) (a, d, F) ] 1 Normal subgroup lattice ----------------------- [3] Order 120 Length 1 Maximal Subgroups: 2 --- [2] Order 60 Length 1 Maximal Subgroups: 1 --- [1] Order 1 Length 1 Maximal Subgroups: G | Cyclic(2) * | Alternating(5) 1 [ Symmetric group S5 acting on a set of cardinality 5 Order = 120 = 2^3 * 3 * 5 (c, b, a, d, F) (c, b), Permutation group acting on a set of cardinality 5 Order = 60 = 2^2 * 3 * 5 (a, d, F) (c, b, a), Permutation group acting on a set of cardinality 5 Order = 1 ] 3 Permutation group acting on a set of cardinality 5 Order = 1 Permutation group acting on a set of cardinality 5 Order = 1 Permutation group acting on a set of cardinality 5 Order = 1 Symmetric group S5 acting on a set of cardinality 5 Order = 120 = 2^3 * 3 * 5 (c, b, a, d, F) (c, b) Mapping from: GrpPerm: S5 to GrpPerm: S5 Permutation group acting on a set of cardinality 5 Order = 1 Character Table of Group S5 --------------------------- ----------------------------- Class | 1 2 3 4 5 6 7 Size | 1 10 15 20 30 24 20 Order | 1 2 2 3 4 5 6 ----------------------------- p = 2 1 1 1 4 3 6 4 p = 3 1 2 3 1 5 6 2 p = 5 1 2 3 4 5 1 7 ----------------------------- X.1 + 1 1 1 1 1 1 1 X.2 + 1 -1 1 1 -1 1 -1 X.3 + 4 2 0 1 0 -1 -1 X.4 + 4 -2 0 1 0 -1 1 X.5 + 5 1 1 -1 -1 0 1 X.6 + 5 -1 1 -1 1 0 -1 X.7 + 6 0 -2 0 0 1 0 [] ( 5, 3, 1, 2, 1, 0, 0 ) A group of automorphisms of GrpPerm: S5, Degree 5, Order 2^3 * 3 * 5 Generators: Automorphism of GrpPerm: S5, Degree 5, Order 2^3 * 3 * 5 which maps: (c, b, a, d, F) |--> (c, F, d, a, b) (c, b) |--> (d, F) Automorphism of GrpPerm: S5, Degree 5, Order 2^3 * 3 * 5 which maps: (c, b, a, d, F) |--> (c, F, a, b, d) (c, b) |--> (b, a) Automorphism of GrpPerm: S5, Degree 5, Order 2^3 * 3 * 5 which maps: (c, b, a, d, F) |--> (c, a, d, F, b) (c, b) |--> (c, b) <17, 5, 2> [ c, b, a, d ]

cjsh

S4的：一般书上只有S3,S4 闲麻烦里面的S5是S4，只有4个基数a,b,c,d, 11个子群，4个正规子群， 4个自同构群， 没6阶子群 S5 := Sym({ "a", "b", "c", "d"}); > S5; Identity(S5); Degree(S5); Generators(S5); GeneratorsSequence(S5); NumberOfGenerators(S5); GSet(S5) ; Order(S5) ; FactoredOrder(S5); IsEven(S5); IsSimple(S5) ; IsSoluble(S5); Exponent(S5) ; NumberingMap(S5) ; Representative(S5) ; NumberOfClasses(S5) ; Centre(S5); SubgroupClasses(S5); NormalSubgroups(S5); SolvableSubgroups(S5); SimpleSubgroups(S5); DerivedSeries(S5); DerivedLength(S5) ; NormalLattice(S5) ; ChiefFactors(S5); ChiefSeries(S5); list := SmallGroups(120:Search:="Insoluble"); > #list; FittingSubgroup(S5) ; FrattiniSubgroup(S5); Radical(S5) ; RadicalQuotient(S5); CharacterTable(S5); PermutationCharacter(S5) ; AutomorphismGroup(S5); NameSimple(S5) ; Base(S5) ; Symmetric group S5 acting on a set of cardinality 4 Order = 24 = 2^3 * 3 Id(S5) 4 { (c, b, a, d), (c, b) } [ (c, b, a, d), (c, b) ] 2 GSet{@ c, b, a, d @} 24 [ <2, 3>, <3, 1> ] false false true 12 Mapping from: GrpPerm: S5 to { 1 .. 24 } Id(S5) 5 Permutation group acting on a set of cardinality 4 Order = 1 Conjugacy classes of subgroups ------------------------------ [ 1] Order 1 Length 1 Permutation group acting on a set of cardinality 4 Order = 1 [ 2] Order 2 Length 3 Permutation group acting on a set of cardinality 4 Order = 2 (c, d)(b, a) [ 3] Order 2 Length 6 Permutation group acting on a set of cardinality 4 Order = 2 (a, d) [ 4] Order 3 Length 4 Permutation group acting on a set of cardinality 4 Order = 3 (b, a, d) [ 5] Order 4 Length 1 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (c, d)(b, a) (c, a)(b, d) [ 6] Order 4 Length 3 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (c, d, b, a) (c, b)(a, d) [ 7] Order 4 Length 3 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (a, d) (c, b)(a, d) [ 8] Order 6 Length 4 Permutation group acting on a set of cardinality 4 Order = 6 = 2 * 3 (a, d) (b, a, d) [ 9] Order 8 Length 3 Permutation group acting on a set of cardinality 4 Order = 8 = 2^3 (a, d) (c, d)(b, a) (c, a)(b, d) [10] Order 12 Length 1 Permutation group acting on a set of cardinality 4 Order = 12 = 2^2 * 3 (b, a, d) (c, d)(b, a) (c, a)(b, d) [11] Order 24 Length 1 Permutation group acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (a, d) (b, a, d) (c, d)(b, a) (c, a)(b, d) Conjugacy classes of subgroups ------------------------------ [1] Order 1 Length 1 Permutation group acting on a set of cardinality 4 Order = 1 [2] Order 4 Length 1 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (c, d)(b, a) (c, a)(b, d) [3] Order 12 Length 1 Permutation group acting on a set of cardinality 4 Order = 12 = 2^2 * 3 (b, a, d) (c, d)(b, a) (c, a)(b, d) [4] Order 24 Length 1 Permutation group acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (a, d) (b, a, d) (c, d)(b, a) (c, a)(b, d) Conjugacy classes of subgroups ------------------------------ [ 1] Order 1 Length 1 Permutation group acting on a set of cardinality 4 Order = 1 [ 2] Order 2 Length 3 Permutation group acting on a set of cardinality 4 Order = 2 (c, d)(b, a) [ 3] Order 2 Length 6 Permutation group acting on a set of cardinality 4 Order = 2 (a, d) [ 4] Order 3 Length 4 Permutation group acting on a set of cardinality 4 Order = 3 (b, a, d) [ 5] Order 4 Length 1 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (c, d)(b, a) (c, a)(b, d) [ 6] Order 4 Length 3 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (c, d, b, a) (c, b)(a, d) [ 7] Order 4 Length 3 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (a, d) (c, b)(a, d) [ 8] Order 6 Length 4 Permutation group acting on a set of cardinality 4 Order = 6 = 2 * 3 (a, d) (b, a, d) [ 9] Order 8 Length 3 Permutation group acting on a set of cardinality 4 Order = 8 = 2^3 (a, d) (c, d)(b, a) (c, a)(b, d) [10] Order 12 Length 1 Permutation group acting on a set of cardinality 4 Order = 12 = 2^2 * 3 (b, a, d) (c, d)(b, a) (c, a)(b, d) [11] Order 24 Length 1 Symmetric group acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (a, d) (b, a, d) (c, d)(b, a) (c, a)(b, d) Conjugacy classes of subgroups ------------------------------ [ Symmetric group S5 acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (c, b, a, d) (c, b), Permutation group acting on a set of cardinality 4 Order = 12 = 2^2 * 3 (c, b, a) (b, a, d), Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (c, d)(b, a) (c, a)(b, d), Permutation group acting on a set of cardinality 4 Order = 1 ] 3 Normal subgroup lattice ----------------------- [4] Order 24 Length 1 Maximal Subgroups: 3 --- [3] Order 12 Length 1 Maximal Subgroups: 2 --- [2] Order 4 Length 1 Maximal Subgroups: 1 --- [1] Order 1 Length 1 Maximal Subgroups: G | Cyclic(2) * | Cyclic(3) * | Cyclic(2) (2 copies) 1 [ Symmetric group S5 acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (c, b, a, d) (c, b), Permutation group acting on a set of cardinality 4 Order = 12 = 2^2 * 3 (c, b, a) (b, a, d), Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (c, d)(b, a) (c, a)(b, d), Permutation group acting on a set of cardinality 4 Order = 1 ] 3 Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (c, b)(a, d) (c, a)(b, d) Permutation group acting on a set of cardinality 4 Order = 1 Symmetric group S5 acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (c, b, a, d) (c, b) Permutation group acting on a set of cardinality 4 Order = 1 Mapping from: GrpPerm: S5 to GrpPerm: $, Degree 4, Order 1 Permutation group acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (c, b, a, d) (c, b) Character Table of Group S5 --------------------------- ----------------------- Class | 1 2 3 4 5 Size | 1 3 6 8 6 Order | 1 2 2 3 4 ----------------------- p = 2 1 1 1 4 2 p = 3 1 2 3 1 5 ----------------------- X.1 + 1 1 1 1 1 X.2 + 1 1 -1 1 -1 X.3 + 2 2 0 -1 0 X.4 + 3 -1 -1 0 1 X.5 + 3 -1 1 0 -1 [] ( 4, 0, 2, 1, 0 ) A group of automorphisms of GrpPerm: S5, Degree 4, Order 2^3 * 3 Generators: Automorphism of GrpPerm: S5, Degree 4, Order 2^3 * 3 which maps: (c, b, a, d) |--> (c, b, d, a) (c, b) |--> (c, b) Automorphism of GrpPerm: S5, Degree 4, Order 2^3 * 3 which maps: (c, b, a, d) |--> (c, a, d, b) (c, b) |--> (c, a) Automorphism of GrpPerm: S5, Degree 4, Order 2^3 * 3 which maps: (c, b, a, d) |--> (c, d, a, b) (c, b) |--> (a, d) Automorphism of GrpPerm: S5, Degree 4, Order 2^3 * 3 which maps: (c, b, a, d) |--> (c, b, a, d) (c, b) |--> (a, d) >> NameSimple(S5) ; ^ Runtime error in ';NameSimple';: Group is not simple [ c, b, a ]

cjsh

[这个贴子最后由cjsh在 2011/07/08 05:45pm 第 3 次编辑]

S4凯莱表：就24*24都显不出正常样：
S4 := Sym({ "a", "b", "c", "d"});
> S4;
f := NumberingMap(S4);
f;
[ [ f(x*y) : y in S4 ] : x in S4 ];
Symmetric group S4 acting on a set of cardinality 4
Order = 24 = 2^3 * 3
Mapping from: GrpPerm: S4 to { 1 .. 24 }
凯莱表：看子群很直观