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FITTING 子群
G的全部幂零正规子群直积为G的特征子群---F(G)
FRATTINI子群是G的全部极大正规子群之交------φ(G):
1
φ(G)是幂零的,G/φ(G)是幂零的
2
子群和φ(G)直积=G,则子群为G
3
P群是初等交换群,则
φ(G)=1
P群/φ(G)是初等交换群
[br][br]-=-=-=-=- 以下内容由 cjsh 在 时添加 -=-=-=-=-
S4 := Sym({ "a", "b", "c", "d" });
> S4;
NormalSubgroups(S4) ;
DerivedSeries(S4) ;
FittingSubgroup(S4);
FrattiniSubgroup(S4);
MaximalSubgroups(S4) ï¼?
Symmetric group S4 acting on a set of cardinality 4
Order = 24 = 2^3 * 3
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)
[
Symmetric group S4 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
]
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
Conjugacy classes of subgroups
------------------------------
[1] Order 6 Length 4
Permutation group acting on a set of cardinality 4
Order = 6 = 2 * 3
(a, d)
(b, a, d)
[2] 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)
[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)[br][br]-=-=-=-=- 以下内容由 cjsh 在 时添加 -=-=-=-=-
NilpotentSubgroups(S4);
------------------------------
[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
(a, d)
(c, b)(a, d)
[7] 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)
[8] 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) |
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