On $\Sigma_t^{σ}$ -closed classes of finite groups

Abstract

All analyzed groups are finite. Let $\sigma = \{ \sigma_i| i \in I\}$ be a partition of the set of all primes $\mathbb{P}$. If $n$ is an integer, then the symbol $\sigma (n)$ denotes a set $\{\sigma_i| \sigma_i \cap \pi (n) \not = \emptyset\}$. 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 $\Sigma^{\sigma}_ t$ -closed provided $\mathfrak{F}$ contains each group $G$ with subgroups $A_1, ... ,A_t \in \mathfrak{F}$ whose indices $| G : A_1| ,..., | G : A_t|$ are pairwise $\sigma$ -coprime. We study $\Sigma_t^{σ}$ -closed classes of finite groups.