  Let $p$ be a prime number, $G$ be a $p$-solvable finite group and $P$ be a
Sylow $p$-subgroup of $G$. We prove that $G$ is $p$-supersolvable if $N_G(P)$
is $p$-supersolvable and if there is a subgroup $H$ of $P$ with $P' \le H \le
\Phi(P)$ such that $H$ is $s$-semipermutable in $G$. As applications, we
simplify the proofs of some known results and also generalize some known
results.
