A Robinson characterization of finite PσT-groups

Abstract

Let σ = {σi|i ∈ I} be some partition of the set of all primes P and let G be a finite group. Then G is said to be σ-full if G has a Hall σi-subgroup for all i. A subgroup A of G is said to be σ-permutable in G provided G is σ-full and A permutes with all Hall σi-subgroups H of G (that is, AH = HA) for all i. We obtain a characterization of finite groups G in which σ-permutability is a transitive relation in G, that is, if K is a σ-permutable subgroup of H and H is a σ-permutable subgroup of G, then K is a σ-permutable subgroup of G.