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 be a group of prime power order where the prime is . The category can be described as follows:
- Its objects are all the subgroups of
- It is a subcategory of the category of all possible injective homomorphisms between these objects, under composition. In other words, for any two subgroups and , the homomorphism set is a subset of the set of all injective homomorphisms from to , and composition of morphisms is composition as injective homomorphisms.
- It contains all inclusion maps. So, if , then the natural inclusion of in is a member of .
- If is a morphism of , the restriction with co-domain is also a morphism of , and so is the inverse of that restriction. In particular, .