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正规子群的补,SCHUR-ZASSENHAUS定理:
一个正规子群K满足:Gcd(|K|,|G:K|)=1,K则在G有补,。且所有这样的补共轭
试了S4的4阶子群,无补
S4 := Sym({ "a", "b", "c", "d" });
> S4;
NormalSubgroups(S4) ;
DerivedSeries(S4) ;
FittingSubgroup(S4);
FrattiniSubgroup(S4);
sub4:=sub ;
sub4;
s:=S4/sub4;
s;
S4 meet sub4;
s meet sub4;
o4:=Order(S4);
o4;
osub4:=Order(sub4);
osub4;
i:=Index(S4, sub4) ;
i;
IsNormal(S4, sub4);
IsSubnormal(S4, sub4);
Gcd (osub4,i);
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
Permutation group sub4 acting on a set of cardinality 4
(c, d)(b, a)
(c, a)(b, d)
Permutation group s acting on a set of cardinality 3
Order = 6 = 2 * 3
(2, 3)
(1, 2)
Permutation group sub4 acting on a set of cardinality 4
Order = 4 = 2^2
(c, d)(b, a)
(c, a)(b, d)
>> s meet sub4;
^
Runtime error in ';meet';: Could not find a covering group
24
4
6
true
true
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