SUBNORMAL GROUP 的补

cjsh

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 2