Abstract
Let F be a class of finite groups and let G be a finite group. Assume that sigma = {sigma(i) : i is an element of I} is a partition of the set of prime numbers P. A set Z of subgroups of G is called a complete Hall F-set of subgroups of G of type s if (i) for every W. Z, W. F and W is a Hall (i)-subgroup of G for some i. I and if (ii) for every si. s, Z contains exactly one and only one Hall si -subgroup of G. A subgroup H of G is said to be Z -permutable in G if H permutes with every member of Z. We investigate the influence of Z -permutable subgroups on the structure of finite groups. Also, we study some properties of Z -permutable subgroups. Several results from the literature are improved and generalized.