# Monomorphism iff injective in the category of groups

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term injective homomorphism

View a complete list of pages giving proofs of equivalence of definitions

## Contents

## Statement

The following are equivalent for a homomorphism of groups :

- is injective as a set map.
- is a monomorphism with respect to the category of groups: For any homomorphisms from any group , .

## Related facts

## Proof

### Injective homomorphism implies monomorphism

This follows simply by thinking of the maps as set maps. In general, for any concrete category, any injective homomorphism is a monomorphism.

### Monomorphism implies injective homomorphism

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**Given**: is a monomorphism: For any homomorphisms from any group , .

**To prove**: is injective, i.e., the kernel of is the trivial subgroup of .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | Let be the kernel of . Let be natural inclusion of in and be the trivial homomorphism from to . | ||||

2 | as homomorphisms from to , since both and are trivial homomorphisms from to . | Step (1) | By Step (1), is the kernel of and is its natural inclusion in , so the composite is trivial. is trivial because is trivial. | ||

3 | . | is a monomorphism | Step (2) | Step-given combination direct. | |

4 | is trivial, i.e., the kernel of is trivial. | Steps (1), (3) | Step (3) tells us that the inclusion of in is the same as the trivial homomorphism, forcing to be trivial. This completes the proof. |