# 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 $Q \le R \le P$, 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, $Q \cong \varphi(Q)$.