A generalization of Kramer’s theory

Abstract

Throughout this paper, all groups are finite. ${\sigma =\{\sigma_{i}\mid i\in I \}}$ is some partition of the set of all primes ${\mathbb{P}}$, and ${\sigma (n)= \{\sigma _{i}\mid \sigma _{i}\cap \pi (n)\ne \emptyset \}}$ for any ${n\in \mathbb{N}}$. The natural numbers 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 ${\Gamma_{t}^{\sigma}}$-closed (respectively weakly${\Gamma_{t}^{\sigma}}$-closed) provided ${\mathfrak{F}}$ contains each finite group G which satisfies the following conditions:

• (1) G has subgroups ${A_{1}, \ldots, A_{t} \in \mathfrak{F}}$ such that G = A_iA_j for all ${i\ne j}$ ;

• (2) The indices ${|G:N_{G}(A_{1})|, \ldots, |G:N_{G}(A_{t})|}$ (respectively the indices ${|{G:A_{1}}|, \ldots, |G:A_{t-1}|, |G:N_{G}(A_{t})|}$) are pairwise ${\sigma}$-coprime.

We study properties and some applications of (weakly) ${\Gamma_{t}^{\sigma}}$-closed classes of finite groups.