# Epimorphism iff surjective in the category of groups

From Groupprops

## Contents

## Statement

The following are equivalent for a homomorphism of groups :

- is surjective as a set map.
- is an epimorphism with respect to the category of groups: For any homomorphisms to any group , .

## Related facts

## Proof

### Surjective homomorphism implies epimorphism

This follows simply by thinking of the maps as set maps. In general, for any concrete category, any surjective homomorphism is an epimorphism.

### Epimorphism implies surjective homomorphism

The idea here is to define as the amalgamated free product of two copies of amalgamated over the image in , and take and as the embeddings of the two copies of respectively in .

With this approach, we may end up with an infinite even if and are finite. There are slight modifications of this approach that can be used to guarantee that is finite whenever is finite.**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]