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

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

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