The Local Structure for Finite Groups with a Large $p$-Subgroup

Let $p$ be a prime, $G$ a finite $mathcal{K}_p$-group $S$ a Sylow $p$-subgroup of $G$ and $Q$ a large subgroup of $G$ in $S$ (i.e., $C_G(Q) leq Q$ and $N_G(U) leq N_G(Q)$ for $1 e U leq C_G(Q)$). Let $L$ be any subgroup of $G$ with $Sleq L$, $O_p(L) eq 1$ and $Q trianglelefteq L$. In this paper the authors determine the action of $L$ on the largest elementary abelian normal $p$-reduced $p$-subgroup $Y_L$ of $L$.