One application of the $\sigma$-local formations of finite groups

Abstract

Throughout this paper, all groups are finite. Let $\sigma =\{\sigma_{i} | i\in I \}$ be some partition of the set of all primes $\Bbb{P}$. If $n$ is an integer, the symbol $\sigma (n)$ denotes the set $\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset \}$. The integers $n$ and $m$ are called $\sigma$-coprime if $\sigma (n)\cap \sigma (m)=\emptyset$. Let $t > 1$ be a natural number and let $\mathfrak{F}$ be a class of groups. Then we say that $\mathfrak{F}$ is $\Sigma_{t}^{\sigma}$-closed provided $\mathfrak{F}$ contains each group $G$ with subgroups $A_{1}, \ldots , A_{t}\in \mathfrak{F}$ whose indices $|G:A_{1}|$, $\ldots$, $|G:A_{t}|$ are pairwise $\sigma$-coprime. In this paper, we study $\Sigma_{t}^{\sigma}$-closed classes of finite groups.