Finite Generalized Soluble Groups

Abstract

Let σ = {σ_i | i ∈ I} be a partition of the set of all primes ℙ and G a finite group. Suppose σ(G) = {σ_i | σ_i ∩ π(G) ≠ = ∅}. A set ℋ of subgroups of G is called a complete Hall σ-set of G if every nontrivial member of ℋ is a σ_i-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σ_i-subgroup of G for every i such that σ_i ∈ σ(G). A group G is σ-full if G possesses a complete Hall σ-set. A complete Hall σ-set ℋ of G is called a σ-basis of G if every two subgroups A, B ∈ ℋ are permutable, i.e., AB = BA. In this paper, we study properties of finite groups having a σ-basis. It is proved that if G has a σ-basis, then G is generalized σ-soluble, i.e, |σ(H/K)| ≤ 2 for every chief factor H/K of G. Moreover, it is shown that every complete Hall σ-set of a σ-full group G forms a σ-basis of G iff G is generalized σ-soluble, and for the automorphism group G/C_G(H/K) induced by G on any its chief factor H/K, we have |σ(G/C_G(H/K))| ≤ 2 and also σ(H/K) ⊆ σ(G/C_G(H/K)) in the case |σ(G/C_G(H/K))| = 2.