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[原创]关于20面体
[这个贴子最后由cjsh在 2011/07/12 05:18pm 第 1 次编辑]
群方程就专解这种问题的---------24阶群同构正6、8面体变换群
类方程:
|G| = |Z(G)| + ∑i [G : Hi]
看下面24阶群轨道稳定子信息,对着书上这类题正6、8面体变换群能很快的出答案,这种问题阶大了只有计算机才能很好的解决
和彩珠串一样这是一大类问题,深了可和组合图论,算法图论连着
Symmetric group S acting on a set of cardinality 4
Order = 24 = 2^3 * 3
Permutation group G acting on a set of cardinality 4
Order = 24 = 2^3 * 3
(1, 2, 3, 4)
(1, 2)
{
(1, 2, 3, 4),
(1, 2)
}
2
GSet{@ 1, 2, 3, 4 @}
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 3
Rep (1, 2)(3, 4)
[3] Order 2 Length 6
Rep (1, 2)
[4] Order 3 Length 8
Rep (1, 2, 3)
[5] Order 4 Length 6
Rep (1, 2, 3, 4)
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 3
Rep (1, 2)(3, 4)
[3] Order 2 Length 6
Rep (1, 2)
[4] Order 3 Length 8
Rep (1, 2, 3)
[5] Order 4 Length 6
Rep (1, 2, 3, 4)
Mapping from: GrpPerm: G to { 1 .. 5 }
5
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
(1, 4)(2, 3)
[ 3] Order 2 Length 6
Permutation group acting on a set of cardinality 4
Order = 2
(3, 4)
[ 4] Order 3 Length 4
Permutation group acting on a set of cardinality 4
Order = 3
(2, 3, 4)
[ 5] Order 4 Length 1
Permutation group acting on a set of cardinality 4
Order = 4 = 2^2
(1, 4)(2, 3)
(1, 3)(2, 4)
[ 6] Order 4 Length 3
Permutation group acting on a set of cardinality 4
Order = 4 = 2^2
(1, 4, 2, 3)
(1, 2)(3, 4)
[ 7] Order 4 Length 3
Permutation group acting on a set of cardinality 4
Order = 4 = 2^2
(3, 4)
(1, 2)(3, 4)
[ 8] Order 6 Length 4
Permutation group acting on a set of cardinality 4
Order = 6 = 2 * 3
(3, 4)
(2, 3, 4)
[ 9] Order 8 Length 3
Permutation group acting on a set of cardinality 4
Order = 8 = 2^3
(3, 4)
(1, 4)(2, 3)
(1, 3)(2, 4)
[10] Order 12 Length 1
Permutation group acting on a set of cardinality 4
Order = 12 = 2^2 * 3
(2, 3, 4)
(1, 4)(2, 3)
(1, 3)(2, 4)
[11] Order 24 Length 1
Permutation group acting on a set of cardinality 4
Order = 24 = 2^3 * 3
(3, 4)
(2, 3, 4)
(1, 4)(2, 3)
(1, 3)(2, 4)
Partially ordered set of subgroup classes
-----------------------------------------
[11] Order 24 Length 1 Maximal Subgroups: 8 9 10
---
[10] Order 12 Length 1 Maximal Subgroups: 4 5
[ 9] Order 8 Length 3 Maximal Subgroups: 5 6 7
---
[ 8] Order 6 Length 4 Maximal Subgroups: 3 4
[ 7] Order 4 Length 3 Maximal Subgroups: 2
[ 6] Order 4 Length 3 Maximal Subgroups: 2 3
[ 5] Order 4 Length 1 Maximal Subgroups: 2
---
[ 4] Order 3 Length 4 Maximal Subgroups: 1
[ 3] Order 2 Length 6 Maximal Subgroups: 1
[ 2] Order 2 Length 3 Maximal Subgroups: 1
---
[ 1] Order 1 Length 1 Maximal Subgroups:
[
GSet{@ 1, 2, 3, 4 @}
]
[ <4, 1> ]
{}
Symmetric group S acting on a set of cardinality 4
Order = 24 = 2^3 * 3
Permutation group G acting on a set of cardinality 4
Order = 24 = 2^3 * 3
(1, 2, 3, 4)
(1, 2)
{
(1, 2, 3, 4),
(1, 2)
}
2
GSet{@ 1, 2, 3, 4 @}
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 3
Rep (1, 2)(3, 4)
[3] Order 2 Length 6
Rep (1, 2)
[4] Order 3 Length 8
Rep (1, 2, 3)
[5] Order 4 Length 6
Rep (1, 2, 3, 4)
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 3
Rep (1, 2)(3, 4)
[3] Order 2 Length 6
Rep (1, 2)
[4] Order 3 Length 8
Rep (1, 2, 3)
[5] Order 4 Length 6
Rep (1, 2, 3, 4)
Mapping from: GrpPerm: G to { 1 .. 5 }
5
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
(1, 4)(2, 3)
[ 3] Order 2 Length 6
Permutation group acting on a set of cardinality 4
Order = 2
(3, 4)
[ 4] Order 3 Length 4
Permutation group acting on a set of cardinality 4
Order = 3
(2, 3, 4)
[ 5] Order 4 Length 1
Permutation group acting on a set of cardinality 4
Order = 4 = 2^2
(1, 4)(2, 3)
(1, 3)(2, 4)
[ 6] Order 4 Length 3
Permutation group acting on a set of cardinality 4
Order = 4 = 2^2
(1, 4, 2, 3)
(1, 2)(3, 4)
[ 7] Order 4 Length 3
Permutation group acting on a set of cardinality 4
Order = 4 = 2^2
(3, 4)
(1, 2)(3, 4)
[ 8] Order 6 Length 4
Permutation group acting on a set of cardinality 4
Order = 6 = 2 * 3
(3, 4)
(2, 3, 4)
[ 9] Order 8 Length 3
Permutation group acting on a set of cardinality 4
Order = 8 = 2^3
(3, 4)
(1, 4)(2, 3)
(1, 3)(2, 4)
[10] Order 12 Length 1
Permutation group acting on a set of cardinality 4
Order = 12 = 2^2 * 3
(2, 3, 4)
(1, 4)(2, 3)
(1, 3)(2, 4)
[11] Order 24 Length 1
Permutation group acting on a set of cardinality 4
Order = 24 = 2^3 * 3
(3, 4)
(2, 3, 4)
(1, 4)(2, 3)
(1, 3)(2, 4)
Partially ordered set of subgroup classes
-----------------------------------------
[11] Order 24 Length 1 Maximal Subgroups: 8 9 10
---
[10] Order 12 Length 1 Maximal Subgroups: 4 5
[ 9] Order 8 Length 3 Maximal Subgroups: 5 6 7
---
[ 8] Order 6 Length 4 Maximal Subgroups: 3 4
[ 7] Order 4 Length 3 Maximal Subgroups: 2
[ 6] Order 4 Length 3 Maximal Subgroups: 2 3
[ 5] Order 4 Length 1 Maximal Subgroups: 2
---
[ 4] Order 3 Length 4 Maximal Subgroups: 1
[ 3] Order 2 Length 6 Maximal Subgroups: 1
[ 2] Order 2 Length 3 Maximal Subgroups: 1
---
[ 1] Order 1 Length 1 Maximal Subgroups:
[
GSet{@ 1, 2, 3, 4 @}
]
[ <4, 1> ]
{}
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