Category on a finite p-group

From Groupprops

This term is related to: fusion systems
See more terms related to fusion systems OR see facts/theorems related to fusion systems

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 $Hom_{\mathcal{F}}(Q,R)$ 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 $Hom_{\mathcal{F}}(Q,R)$