Category of groups

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In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.

The monomorphisms in Grp are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorphisms are precisely the bijective homomorphisms.

The category Grp is both complete and co-complete. The category-theoretical product in Grp is just the direct product of groups while the category-theoretical coproduct in Grp is the free product of groups. The zero objects in Grp are the trivial groups (consisting of just an identity element).

The category of abelian groups, Ab, is a full subcategory of Grp. It is an abelian category, but Grp is not. Indeed, Grp isn't even an additive category, because there is no natural way to define the "sum" of two group homomorphisms.

Every morphism f : G → H in Grp has a category-theoretical kernel (given by the ordinary kernel of algebra ker f = {x in G | f(x) = e}), and also a category-theoretical cokernel (given by the factor group of H by the normal closure of f(H) in H). Unlike in abelian categories, it is not true that every monomorphism in Grp is the kernel of its cokernel.

The notion of exact sequence is meaningful in Grp, and some results from the theory of abelian categories, such as the nine lemma, the five lemma, and their consequences hold true in Grp. The snake lemma however is not true in Grp.