Category of Groups

The Category of groups and group homomorphisms

Lemma

Category is Balanced.

Proof

Similarly as in Category of Sets: Injective homomorphisms are Monomorphisms and surjective homomorphisms are Epimorphisms. To see the other way, let be a Monomorphism. Suppose Consider with and . Then , as is Monomorphism.

Now suppose that is an Epimorphism. Let , the set of right Cosets and let (we just extend with something that’s not contained in ) Let be the Homomorphism induced by the Group action of on which fixes . Consider a permutation that exchanges and and let such that .

First note that if , then fixes . Thus commutes with and . We conclude that as is Epimorphism.

But then has to fix for all , which means that i.e. is surjective.