Finite groups with weakly m-σ-permutable subgroups

Abstract

Let [Formula: see text] be a finite group, [Formula: see text] be a partition of the set of all primes [Formula: see text] 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 [Formula: see text] if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] possesses a complete Hall [Formula: see text]-set [Formula: see text] such that [Formula: see text] for all [Formula: see text] and all [Formula: see text]. Let [Formula: see text] be a subgroup of [Formula: see text]. [Formula: see text] is: [Formula: see text]-[Formula: see text]-permutable in [Formula: see text] if [Formula: see text] for some modular subgroup [Formula: see text] and [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text]; weakly[Formula: see text]-[Formula: see text]-permutable in [Formula: see text] if there are an [Formula: see text]-[Formula: see text]-permutable subgroup [Formula: see text] and a [Formula: see text]-subnormal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we investigate the influence of weakly [Formula: see text]-[Formula: see text]-permutable subgroups on the structure of finite groups.