Category on a finite p-group

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Definition

A category on a finite p-group is a category defined relative to a group of prime power order as follows. Let $P$ be a group of prime power order where the prime is $p$ . The category ${\mathcal {F}}$ can be described as follows:

Its objects are all the subgroups of $P$
It is a subcategory of the category of all possible injective homomorphisms between these objects, under composition. In other words, for any two subgroups $Q$ and $R$ , the homomorphism set $\operatorname {Hom} _{\mathcal {F}}(Q,R)$ is a subset of the set of all injective homomorphisms from $Q$ to $R$ , and composition of morphisms is composition as injective homomorphisms.
It contains all inclusion maps. So, if $Q\leq R\leq P$ , then the natural inclusion of $Q$ in $R$ is a member of $\operatorname {Hom} _{\mathcal {F}}(Q,R)$ .
If $\varphi :Q\to R$ is a morphism of ${\mathcal {F}}$ , the restriction with co-domain $\varphi (Q)$ is also a morphism of ${\mathcal {F}}$ , and so is the inverse of that restriction. In particular, $Q\cong \varphi (Q)$ .

References

Introduction to Fusion Systems by Markus Linckelmann^Weblink^{More info}