Abstract:
Consider a finite group $G$. A subgroup is called $S$-quasinormal whenever it permutes with all Sylow subgroups of $G$. Denote by $B_{sG}$ the largest $S$-quasinormal subgroup of $G$ lying in $B$. A subgroup $B$ is called $S$-supplemented in $G$ whenever there is a subgroup $T$ with $G=BT$ and $B\cap T\le B_{sG}$. A subgroup $L$ of $G$ is called a quaternionic subgroup whenever $G$ has a section $A/B$ isomorphic to the order 8 quaternion group such that $L\le A$ and $L\cap B=1$. This article is devoted to proving the following theorem.
Theorem. Let $E$ be a normal subgroup of a group $G$ and let $p$ be a prime divisor of $|E|$ such that $(p-1,|E|)=1$. Take a Sylow $p$-subgroup $P$ of $E$. Suppose that either all maximal subgroups of $P$ lacking $p$-supersoluble supplement in $G$ or all order $p$ subgroups and quaternionic order 4 subgroups of $P$ lacking $p$-supersoluble supplement in $G$ are $S$-supplemented in $G$. Then $E$ is $p$-nilpotent and all its $G$-chief $p$-factors are cyclic.
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