A result on s-semipermutable subgroups of finite groups and some applications

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 $P^{\mathrm{\prime }}\le H\le \mathrm{\Phi }(P)$ such that $H$ is $s$-semipermutable in $G$. As applications, we simplify the proofs of some known results and also generalize some known results.