次正规嵌入子群与有限群的可解性(Ⅱ)

Subnormally Embedded Subgroups and Solvability of Finite Groups

群G可解当且仅当对于每个M∈Fod(G)或M∈F2(G)或存在G的可解极大子群M,存在I(M)的极大元C使得C/K(C)幂零且下列条件之一得到满足:(1)C/K(C)的Sylow 2-子群的极大子群在G/K(C)中次正规嵌入;(2)C/K(C)的Sylow 2-子群的循环子群在G/K(C)中次正规嵌入.

G is solvable if and only for each M∈Fod (G)or existing solvable maximal subgroup M in G, there is a maximal element C in I(M)such that C/K(C)is nilpotent and one of the following conditions is contented.(1)A maximal subgroup of Sylow 2-subgroup of C/K(C)is subnormally embedded in G;(2)A cyclic subgroup of Sylow 2-subgroup of C/K(C)is subnormally embedded in G.