A new characterization of finite σ-soluble PσT-groups - Adarchenko, N.M. (Algebra and Discrete Mathematics, 2020)
Let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and G a finite group. G is said to be σ-soluble if every chief factor H/K of G is a σᵢ-group for some i = i(H/K). 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 σᵢ-subgroup of G for some σᵢ∈ σ and H contains exactly one Hall σᵢ-subgroup of G for every i such that σᵢ ∩ π(G) ≠ ∅. A subgroup A of G is said to be σ-quasinormal or σ-permutable in G if G has a complete Hall σ-set H such that AHˣ = HˣA for all x ∈ G and all H ∈ H. We obtain a new characterization of finite σ-soluble groups G in which σ-permutability is a transitive relation in G.