In my previous post, we saw that injections (surjections) in concrete categories are always monomorphisms (epimorphisms), and in some cases the converse holds.
We now wish to classify all epimorphisms of groups. To show that all epimorphisms are surjective, for any strict subgroup we want to construct maps to some group that differ on but agree on . In the case of abelian groups this is relatively easy, because we can take to be the cokernel, the quotient map, and the zero map. But in general the cokernel only exists if the image is normal, so a different argument is needed.
Lemma. Let be a group homomorphism. Then is an epimorphism if and only if is surjective.
Proof. We already saw that surjections are epimorphisms. Conversely, let be an epimorphism of groups. We may replace by its image in , since the map is still an epimorphism. Let be the coset space, viewed as a pointed set with distinguished element . Let be the set “ with the distinguished point doubled”, and write and for these distinguished points.
Let be the symmetric group on , and define homomorphisms by letting act naturally on the copy of in (for ). Since the action of on fixes the trivial coset , we see that the maps agree. Since is an epimorphism, this forces . But then
showing that is surjective (and a fortiori ).
Note however that the result is not true in every algebraic category. For example, the map is an epimorphism of (commutative) rings that is not surjective. More generally, every localisation is an epimorphism, by the universal property of localisation; these maps are rarely surjective.