On an open problem of Guo-Skiba

Abstract

Let σ = {σ i |i ∈ I} be some partition of the set P of all primes, that is, P = i∈I σ i and σ i ∩ σ j = ∅ for all i = j. 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 non-identity member of H is a Hall σ i -subgroup of G and H contains exactly one Hall σ i -subgroup of G for every σ i ∈ σ(G). G is said to be a σ-group if it possesses a complete Hall σ-set. A σ-group G is said to be σ-dispersive provided G has a normal series 1 = G 1 < G 2 < · · · < G t < G t+1 = G and a complete Hall σ-set {H 1 , H 2 , · · · , H t } such that G i H i = G i+1 for all i = 1, 2, . . . t. In this paper, we give a characterizations of σ-dispersive group, which give a positive answer to an open problem of Skiba in the paper [1].