On weakly σ-semipermutable subgroups of finite groups

Abstract

Let [Formula: see text] be some partition of the set of all primes [Formula: see text], G be a finite group and [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall [Formula: see text]-set of G if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of G for some i and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of G for every [Formula: see text]. Let [Formula: see text] be a complete Hall [Formula: see text]-set of G. A subgroup A of G is said to be [Formula: see text]-semipermutable with respect to [Formula: see text] if [Formula: see text] for all [Formula: see text] and all [Formula: see text] such that [Formula: see text]; [Formula: see text]-semipermutable in G if A is [Formula: see text]-semipermutable in G with respect to some complete Hall [Formula: see text]-set of G. We say that a subgroup H of G is weakly [Formula: see text]-semipermutable in G if there exists a [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text] such that HT is [Formula: see text]-permutable in G and [Formula: see text], where [Formula: see text] is the subgroup of H generated by all those subgroups of H which are [Formula: see text]-semipermutable in G. In this paper, we study the structure of G under the condition that some subgroups of G are weakly [Formula: see text]-semipermutable in G.