Fusion system

From Groupprops

This is a variation of group
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Definition

Let $P$ be a group of prime power order, say a finite $p$ -group, for a prime $p$ . A fusion system $\mathcal{F}$ on $P$ is a category on $P$ with the following properties:

For any subgroups $Q, R \le P$ , all injective homomorphisms from $Q$ to $R$ that arise as restrictions of inner automorphisms of $P$ , are present in $\mathcal{F}$
It satisfies the extension axiom. The extension axiom is as follows:

Call a subgroup $R$ of $P$ fully normalized by $\mathcal{F}$ if $|N_P(R)| \le |N_P(Q)|$ for any $Q \cong R$ where $\cong$ is isomorphism in the category $\mathcal{F}$ .