Characterizations of finite groups with σ-semiembedded subgroups

Abstract

Let [Formula: see text] be a partition of the set of all primes [Formula: see text], [Formula: see text] 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 [Formula: see text] if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] for some [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] has 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]; [Formula: see text]-semipermutable in [Formula: see text] if [Formula: see text] has 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]-group [Formula: see text] with [Formula: see text]. We say that [Formula: see text] is [Formula: see text]-semiembedded in [Formula: see text] if there exists a [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a [Formula: see text]-permutable subgroup of [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the subgroup of [Formula: see text] which is generated by all [Formula: see text]-semipermutable subgroups of [Formula: see text] contained in [Formula: see text]. In this paper, we study the influence of [Formula: see text]-semiembedded subgroups on the structure of finite groups. Some known results are generalized and unified.