Local definitions of formations of finite groups

TL;DR

This paper investigates local definitions of formations in finite groups, providing new proofs and establishing conditions under which formations have specific local and satellite properties.

Contribution

It introduces a new proof for the existence of an $ ext{omega}$-composition satellite and links local definitions to formation properties in finite groups.

Findings

Proved the existence of an $ ext{omega}$-composition satellite for $ ext{omega}$-solubly saturated formations.

Showed that $ ext{X}$-local formations have an $ ext{X}$-composition satellite.

Analyzed relations between local definitions of various formation types.

Abstract

A problem of constructing of local definitions for formations of finite groups is discussed in the article. The author analyzes relations between local definitions of various types. A new proof of existence of an $\omega$-composition satellite of an $\omega$-solubly saturated formation is obtained. It is proved that if a non-empty formation of finite groups is $\mathfrak X$-local by F\"{o}rster, then it has an $\mathfrak X$-composition satellite.