弱S-正规子群

WEAKLY S-NORMAL SUBGROUPS

本文定义了弱 S-正规子群的概念,并得到利用这一概念来刻划 Sylow 塔群、超可解群,(?)群和(?)群的几个结果。

A subgroup H of a finite group G is called a weakly S-normal subgroupof G if there is a Sylow p-subgroup Sp of G such that HS_p=S_pH for eachprime P||G|.In this paper we present characterization of some classes ofgroups through weakly S-normal subgroups.Theorem 1.A finite group G is (?)-group if and only if each primitvesubgroup of G is weakly S-normal subgroup of G.Theorem 2.A finte group G is (?)-group if and only if each subgroup of Gis weakly S-normal subgroup of G.Theorem 3 A finite group G is supersolvable if and only if each subgr-oup H≤G contains a weakly S-normal subroup of order d for each divisor dof |H|.Theorem 4 A finite group G is (?)-group if and only ifi)the nilpotent residual r_∞(G)of G is a nilpotent Hall subgroup;ii)for all H≤r_∞(G),or H is a weakly S-normal snbgroup of G,orN_G(H)=M_1N(M_1,N is respectively π′,π-Hall subgroup of N_G(H) whereπ=π(r_∞(G)),M_1H is not a primitive subgroup of N_G(H).