One application of the σ-local formations of finite groups

Abstract

Throughout this paper, all groups are finite. Let σ = {σi|i ∈ I} be some partition of the set of all primes P. If n is an integer, the symbol σ(n) denotes the set {σi|σi ∩ π(n) 6= ∅}. The integers n and m are called σ-coprime if σ(n) ∩ σ(m) = ∅. Let t > 1 be a natural number and let F be a class of groups. Then we say that F is Σσ t -closed provided F contains each group G with subgroups A1, . . . , At ∈ F whose indices |G : A1|, . . ., |G : At| are pairwise σ-coprime. In this paper, we study Σσ t -closed classes of finite groups.