Some characterizations of finite σ-soluble PσT-groups

Let sigma = {sigma(i) broken vertical bar i is an element of I} be some partition of the set of all primes P and G a finite group. G is said to be sigma-soluble if every chief factor H/K of G is a sigma i-group for some i = i(H/K). A set H of subgroups of G is said to be a complete Hall sigma-set of G if every member not equal 1 of H is a Hall sigma(i)-subgroup of G for some sigma(i) is an element of sigma and H contains exactly one Hall sigma(i)-subgroup of G for every i is an element of I such that sigma(i) boolean AND pi(G) not equal (sic). A subgroup A of G is said to be sigma-permutable in G if G has a complete Hall sigma-set H such that AH(x) = X-x for all x is an element of G and all H is an element of H. We obtain characterizations of finite sigma-soluble groups G in which sigma-permutability is a transitive relation in G. (C) 2017 Elsevier Inc. All rights reserved.