Endomorphism of a group

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).

Relation with other properties

Weaker properties

• Semiendomorphism of a group
• Quasiendomorphism of a group
• 1-endomorphism of a group

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.