Exoticity index

Definition
Suppose $$p$$ is a prime number, $$P$$ is a finite p-group, and $$\mathcal{F}$$ is a saturated fusion system on $$P$$. The exoticity index of $$\mathcal{F}$$ is defined as the smallest of all possible numbers $$\log_p [S:P]$$ (the base $$p$$ logarithm of the index of $$P$$ in $$S$$) where:

$$S$$ is a $$p$$-Sylow subgroup containing $$P$$ in a finite group $$G$$ containing $$P$$ such that $$\mathcal{F}$$ equals the fusion system induced by $$G$$ on $$P$$.

Facts

 * The exoticity index is finite. This follows from the fact that every fusion system on a finite p-group is induced by a finite group containing it. We're thus taking the minimum over a non-empty subset of the nonnegative integers, so it must exist and be finite.
 * The exoticity index is positive if and only if $$\mathcal{F}$$ is an exotic fusion system.