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

Abstract

Let $\mathfrak{F}$ 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 $\mathfrak{F}$ if $G\in \mathfrak{F}$ whenever $\Sigma \subseteq \mathfrak{F}$. 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.

8 pages