# Groups form a full subcategory of semigroups

## Contents

## Statement

The category of groups forms a full subcategory of the category of semigroups, via the forgetful functor that sends a group to its underlying semigroup (*forgetting* the identity element and inverse map).

In other words, the forgetful functor from groups to semigroups, that sends a group to its underlying semigroup, is full, faithful, and injective.

## Related facts

## Facts used

## Proof

### Injectivity

This follows from Fact (1), which says that that the identity element and inverse map in a group are *completely determined* by the binary operation.

### Faithfulness

This follows from the fact that a homomorphism of groups is completely described by what it does at the set level, and hence is completely described by the corresponding homomorphism of semigroups.

### Fullness

The follows from Fact (2), which says that, to check that a map of groups is a homomorphism, it suffices to check that it preserves the group multiplication.