Saturated fusion system

This is a variation of group|Find other variations of group | Read a survey article on varying group

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 $\mathcal{F}$.

Also define, for any morphism $\phi :Q\to P$ in $\mathcal{F}$:

$N_{\phi }=\{y\in N_P(Q)\mid \mathrm{\exists }z\in N_P(\phi (Q)),\phi (yu{y}^{-1})=z\phi (u)z^{-1}\mathrm{\forall }u\in Q\}$

Then the statement of the extension axiom is:

Every morphism $\phi :Q\to P$ such that $\phi (Q)$ is fully $\mathcal{F}$-normalized, extends to a morphism $\psi :N_{\phi }\to P$.

References

• Introduction to Fusion Systems by Markus Linckelmann^{WeblinkMore info}