# Endomorphism of a group

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This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article defines a function property, viz a property of functions from a group to itself

Endomorphism redirects here. For the more general notion, refer endomorphism of a universal algebra

## Definition

### Symbol-free definition

An endomorphism of a group is a homomorphism from the group to itself.

### Definition with symbols

Let $G$ be a group. A map $\sigma$ from $G$ to itself is termed an endomorphism of $G$ if it satisfies all of the following conditions:

• $\sigma(gh) = \sigma(g)\sigma(h)$ whenever $g$ and $h$ are both in $G$
• $\sigma(e) = e$
• $\sigma(g^{-1}) = \sigma(g)^{-1}$

Actually, the second and third condition follow from the first (refer equivalence of definitions of group homomorphism).

## Facts

### Composition

The composite of two endomorphisms of a group is again an endomorphism of the group. This follwos from the fact that the composite of any two isomorphisms is an isomorphism.

### Identity map

The identity map is always an endomorphism.

### Monoid structure

Combining the fact that endomorphisms are closed under composition, and the fact that the identity map is an endomorphism, the endomorphisms of a group form a submonoid of the monoid of all functions from the group to itself. This submonoid is termed the endomorphism monoid. Its invertible elements are precisely automorphisms of the group.