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.