# A generalization of Kramer’s theory

• Z. Chi
• A. N. Skiba
Article

## 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. (1)

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

2. (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.

## Key words and phrases

finite group formation $${\sigma}$$-function $${\sigma}$$-local formation (weakly) $${\Gamma_{t}^{\sigma}}$$-closed class of groups $${\sigma}$$-soluble group $${\sigma}$$-nilpotent group

## Mathematics Subject Classification

20D10 20D15 20D20

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