On weakly s-semipermutable subgroups of finite groups

Abstract

Suppose that G is a finite group and H is a subgroup of G. We say that H is s-semipermutable in G if HGp=GpH for any Sylow p-subgroup Gp of G with (p,|H|)=1; H is weakly s-semipermutable in G if there are a subnormal subgroup T of G and an s-semipermutable subgroup HssG in G contained in H such that G=HT and H∩T⩽HssG. The structure of a finite group with some weakly s-semipermutable subgroups is investigated. Mainly, we get the following local versionʼs result which is a uniform extension of many recent results in literature: Main TheoremAssume that p is a fixed prime inπ(G)and E is a normal subgroup of G andZUϕ(G)denotes the product of all normal subgroups H of G such that all non-Frattini p–G-chief factors of H have order p. ThenE⩽ZUpϕ(G)if there exists a normal subgroup X of G such thatFp⁎(E)⩽X⩽E, whereFp⁎(E)is the generalized p-Fitting subgroup of E, and X satisfies the following: for any Sylow p-subgroup P of X, P has a subgroup D such that1<|D|<|P|and all subgroups H of P with order|H|=|D|and all cyclic subgroups of P with order 4 (if P is a non-abelian 2-group and|D|=2) are weakly s-semipermutable in G.