A note on the embedding properties of $p$-subgroups in finite groups

Abstract

In this note, we prove that a finite group $G$ is $p$-supersolvable if and only if there exists a power $d$ of $p$ with $p^2 \leq d < |P|$ such that $H\cap O^p(G^*_p)$ is normal in $O^p(G)$ for all non-cyclic normal subgroups $H$ of $P$ with $|H| = d$, where $P$ is a Sylow $p$-subgroup of $G$. Moreover, we also prove that either $l_p(G)\leq 1$ and $r_p(G) \leq 2$ or else $|P\cap O^p(G)| > d$ if there exists a power $d$ of $p$ with $1 \leq d < |P|$ such that $H\cap O^p(G^*_{p^2})$ is normal in $O^p(G)$ for all non-meta-cyclic normal subgroups $H$ of $P$ with $|H| = d$.