Finite σ-soluble groups in which σ-permutability is a transitive relation

Let sigma = {sigma(i) vertical bar i is an element of I} be some partition of the set of all primes P, G a finite group and sigma(G) = {sigma(i) vertical bar sigma(i) boolean AND pi p(G) not equal empty set}. A set H of subgroups of G is said to be a complete Hall s-set of G if every member not equal 1 of His 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 sigma(i) is an element of sigma(G). A subgroup A of G is said to be sigma-quasinormal or sigma-permutable in G provided there is a complete Hall sigma-set H of G such that AH(x) = H(x)A for all x is an element of G and all H is an element of H. In this paper, we study the structure of some generalized soluble groups G in which sigma-permutability is a transitive relation in G.