/sci/ - Science & Math

>>11755324
Okay, so just to make sure you know that this isn't a deep exposition of any sort: I haven't used them for anything myself, so I only know the basics. Anyway, the idea is that you take a finite group [math]G[/math], the order of which is divisible by your favourite prime [math]p[/math], and then let [math]P[/math] be a Sylow [math]p[/math]-subgroup. We know that every [math]p[/math]-subgroup is contained by one of the conjugates of [math]P[/math], so we get pretty much all the [math]p[/math]-data (can't think of any better word here) by looking at any one of the Sylows.

We then define a category whose objects are the subgroups of [math]P[/math], and whose morphisms are supposed to reflect the idea of conjugating the elements. To achieve this goal, first restrict to a subset of injective homomorphisms between the subgroups that contains all the conjugation isomorphisms. Since we can restrict to the image, an injective homomorphism induces an isomorphism, and so we want both the induced isomorphism be in the set of morphisms. Furthermore, we want inclusions to be there, as well as the inverse of every isomorphism in the set. The composition is just the composition of group homomorphisms. We can have many examples, the easy ones being the one with all injective homomorphisms as its morphisms, and the one with only the conjugations (this works because the identity is essentially just conjugation with the identity element, and so you actually have a category).
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