Characterizations of some classes of finite σ-soluble PσT -groups

Abstract

Let σ = {σi|i ∈ I} be some partition of the set of all primes P and G a finite group. G is said to be σ-soluble if every chief factor H/K of G is a σi-group for some i = i(H/K). A set H of subgroups of G is said to be a complete Hall σ-set of G if every member 6= 1 of H is a Hall σi-subgroup of G for some σi ∈ σ and H contains exactly one Hall σi-subgroup of G for every i ∈ I such that σi ∩ π(G) 6= ∅. A subgroup A of G is said to be σ-permutable in G if G has a complete Hall σ-set H such that AHx = HxA for all x ∈ G and all H ∈ H. We obtain characterizations of finite σ-soluble groups G in which σ-permutability is a transitive relation in G.