A $G$-covering subgroup system of a finite group for some classes of $\sigma$-soluble groups

TL;DR

The paper introduces a new concept of $G$-covering subgroup systems for classes of finite groups and proves conditions under which certain subgroup sets cover classes like $\sigma$-soluble and $\sigma$-nilpotent groups, answering open questions.

Contribution

It defines $G$-covering subgroup systems and establishes criteria involving supplements to maximal subgroups that characterize classes of $\sigma$-soluble and $\sigma$-nilpotent groups, advancing group theory.

Findings

Subgroup sets with supplements to maximal subgroups form $G$-covering systems.

Results apply to $\sigma$-soluble, $\sigma$-nilpotent, and $P \sigma T$-groups.

Answers to Kourovka notebook questions 19.87 and 19.88.

Abstract

Let ${\frak F}$ be a class of group and $G$ a finite group. Then a set $\Sigma $ of subgroups of $G$ is called a \emph{$G$-covering subgroup system} for the class ${\frak F}$ if $G\in {\frak F}$ whenever $\Sigma \subseteq {\frak F}$. We prove that: {\sl If a set of subgroups $\Sigma$ of $G$ contains at least one supplement to each maximal subgroup of every Sylow subgroup of $G$, then $\Sigma$ is a $G$-covering subgroup system for the classes of all $\sigma$-soluble and all $\sigma$-nilpotent groups, and for the class of all $\sigma$-soluble $P\sigma T$-groups.} This result gives positive answers to questions 19.87 and 19.88 from the Kourovka notebook.