Influence of $\mathscr M_p$-supplemented subgroups on the structure of $p$-modular subgroups

A subgroup $K$ of $G$ is $\mathscr M_p$-supplemented in $G$ if there exists a subgroup $B$ of $G$ such that $G=KB$ and $TB<G$ for every maximal subgroup $T$ of $K$ with $|K:T|=p^\alpha$. We study the structure of the chief factor of $G$ by using $\mathscr M_p$-supplemented subgroups and generalize the results of Monakhov and Shnyparkov by involving the relevant results about the $p$-modular subgroup $O^p(G)$ of $G$.