Finite Groups with a Solvable Hall σ-Basis

Let σ = \σᵢ i ∈ I \ be a partition of the set P of all primes, and let G be a finite group. A set H of subgroups of G is called a complete Hall σ -set of G if every subgroup in H is a σᵢ -Hall subgroup of G for every i ∈ I and H contains exactly one σᵢ -Hall subgroup for every i such that σᵢ ∩ π (G) ≠ . In this paper, we study the structure of the group G ∈ i ∈ IDσᵢ(S) under the condition that all subgroups in every complete Hall σ -set of the group G are permutable.