# Groups form a full subcategory of monoids

## Statement

The category of groups forms a full subcategory of the category of monoids, via the functor that sends a group to its underlying monoids *forgetting* the inverse map.

In other words, the forgetful functor from the category of groups to the category of monoids is full, faithful and injective.

## Related facts

- Groups form a full subcategory of semigroups: This is actually a somewhat stronger fact.
- Monoids do not form a full subcategory of semigroups

## Proof

### Injectivity

This follows from the fact that the inverse map on a group is completely determined by its binary operation and identity element. `Further information: Equivalence of definitions of group`

### Faithfulness

This follows from the fact that a group homomorphism is completely described by what it does at the set level, hence, it is completely described by the corresponding monoid homomorphism.

### Fullness

This follows from the fact that, to check that a set-theoretic map of groups is a group homomorphism, it suffices to check that it preserves the binary operation.