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.