Some Results on the ${\mathcal F}{\Phi}$-Hypercenter of Finite Groups

A subgroup $H$ of $G$ is said to be $\mathcal{M}_p$-supplemented in $G$ if there exists a subgroup $B$ of $G$ such that $G=HB$ and $TB<G$ for every maximal subgroup $T$ of $H$ with $|H:T|=p^{\alpha}$. In this paper, we will investigate the structure of $Z_{\mathcal{F}}{\Phi}(G)$ by using $\mathcal{M}_p$-supplemented subgroups.