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.