Suppose $\mathcal{ E}$ is a normal subsystem of a saturated fusion system $\mathcal{ F}$ over $S$. If $X\leq S$ is fully $\mathcal{ F}$-normalized, then Aschbacher defined a normal subsystem $N_{ \mathcal{ E}}(X)$ of $N_{ \mathcal{ F}}(X)$. In this short note we revisit and generalize this result using the theory of localities. Our more general approach leads in particular to a normal subsystem $C_{ \mathcal{ E}}(X)$ of $C_{ \mathcal{ F}}(X)$ for every $X\leq S$ which is fully $\mathcal{ F}$-centralized.
Categories: math.GR