О Σσᵼ -замкнутых классах конечных групп

Abstract

All analyzed groups are finite. Let σ = {σᵢ | i ∈ I} be a partition of the set of all primes ℙ. If n is an integer, then the symbol σ(n) denotes a set {σᵢ|σᵢ ∩ π (n) ≠⌀}. 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 Σσᵼ -closed provided 𝔉 contains each group G with subgroups A1, . . . , At ∈ 𝔉 whose indices |G : A₁| , . . . , | G : At| are pairwise σ -coprime. We study Σσᵼ -closed classes of finite groups.