# Endomorphism monoid

## Definition

The **endomorphism monoid** of a group is the monoid defined as follows:

- Its elements are the endomorphisms of the group.
- Its identity element is the identity map from the group to itself.
- Multiplication is by function composition.

The elements of the endomorphism monoid with two-sided inverses form a subgroup of the endomorphism monoid and this subgroup is the automorphism group of the original group.