Finite groups with permutable Hall subgroups.

Let σ = { σ | i ∈ I} be a partition of the set of all primes P, and let G be a finite group. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σ -subgroup of G for some i ∈ I and H contains exactly one Hall σ -subgroup of G for every i such that σ ∩ π( G) ≠ ∅. In this paper, we study the structure of G under the assuming that some subgroups of G permutes with all members of H. [ABSTRACT FROM AUTHOR]