FITTING 子群和FRATTINI子群

cjsh

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)