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

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 $H\cap O^pinsert ignore into journalissuearticles values(G^*_p);$ is normal in $O^pinsert ignore into journalissuearticles values(G);$ 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 $l_pinsert ignore into journalissuearticles values(G);\leq 1$ and $r_pinsert ignore into journalissuearticles values(G); \leq 2$ or else $|P\cap O^pinsert ignore into journalissuearticles values(G);| > d$ if there exists a power $d$ of $p$ with $1 \leq d < |P|$ such that $H\cap O^pinsert ignore into journalissuearticles values(G^*_{p^2});$ is normal in $O^pinsert ignore into journalissuearticles values(G);$ for all non-meta-cyclic normal subgroups $H$ of $P$ with $|H| = d$.