On Cover-Avoiding Subgroups of Sylow Subgroups of Finite Groups

Abstract

A subgroup $H$ of a finite group $G$ is called to have the cover-avoidance property in $G$, $H$ is a CAP subgroup of $G$ in short, if $H$ either covers or avoids every chief factor of $G$. In the present work, we fix a subgroup $D$ in every Sylow subgroup $P$ of $F^{\ast }(G)$ satisfying $1<|D|<|P_{|}$ and study the structure of $G$ under the assumption that every subgroup $H$ with $|H|=|D|$ has the cover-avoidance property in $G$. We state our results in the broader context of formation theory.