# Radical subgroup for a fusion system

Suppose $P$ is a group of prime power order and $\mathcal{F}$ is a fusion system on $P$. A subgroup $Q$ of $P$ is termed a $\mathcal{F}$-radical subgroup if $O_p(\operatorname{Aut}_{\mathcal{F}}(Q)) = \operatorname{Aut}_Q(Q)$. Here, $\operatorname{Aut}_{\mathcal{F}}(Q)$ denotes the automorphisms of $Q$ in the category $\mathcal{F}$, $O_p$ denotes the p-core, and $\operatorname{Aut}_Q(Q)$ denotes the inner automorphism group of $Q$, viewed as a subgroup of $\operatorname{Aut}_{\mathcal{F}}(Q)$.