  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.
