Category on a finite p-group

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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).