On τσ-quasinormal subgroups of finite groups

Let sigma = {sigma(i) vertical bar i is an element of I} be a partition of the set of all primes P and G a finite group. 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 i is an element of I and H contains exactly one Hall sigma(i) - subgroup of G for every i such that sigma(i) boolean AND pi(G) not equal 0. Let tau H(A) = {sigma(i) is an element of sigma(G)\sigma(A)vertical bar sigma (A) vertical bar sigma(A)boolean AND sigma(H-G)not equal 0 for a Hall sigma(i) - subgroup H of G}. We say that a subgroup A of G is tau(sigma)-permutable or tau(sigma)-quasinormal in G with respect to H if AH(chi) = H-chi A for all chi is an element of G and all H is an element of H such that sigma(H) subset of tau(H)(A), and tau(sigma)-permutable or tau(sigma)-quasinormal in G if A is tau(sigma) -permutable in G with respect to some complete Hall sigma-set of G. We study G assuming that tau(sigma)-quasinormality is a transitive relation in G.