On Π-permutable subgroups of finite groups

Abstract

Let σ = {σi|i ∈ I} be some partition of the set of all primes P and Π a non-empty subset of the set σ. A set H of subgroups of a finite group G is said to be a complete Hall Π-set of G if every member of H is a Hall σi-subgroup of G for some σi ∈ Π and H contains exact one Hall σi-subgroup of G for every σi ∈ Π such that σi ∩ π(G) 6= ∅. A subgroup H of G is called Π-quasinormal or Π-permutable in G if G possesses a complete Hall Π-set H = {H1, . . . , Ht} such that AHx i = Hi xA for any i and all x ∈ G. We study the embedding properties of H under the hypothesis that H is Π-permutable in G. Some known results are generalized.