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 $\mathbb{P}$. If $n$ is an integer, the symbol $\sigma (n)$ denotes the set $\{{\sigma }_i|{\sigma }_i\cap \pi (n)\ne \varnothing \}$. The integers $n$ and $m$ are called $\sigma$-coprime if $\sigma (n)\cap \sigma (m)=\varnothing$. 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 $A_1,\dots ,A_t\in \mathfrak{F}$ whose indices $|G:A_1|$, $\dots$, $|G:A_t|$ are pairwise $\sigma$-coprime. In this paper, we study ${\Sigma }_t^{\sigma }$-closed classes of finite groups.

12 pages