A group homomorphism is a function between groups which "respects the group structure".

Definition

Formally, given two groups and $(H, *)$ (which hereafter we will abbreviate as $G$ and $H$ respectively), a group homomorphism from $G$ to $H$ is a function $f$ from the underlying set $G$ to the underlying set $H$ , such that $f (a) * f (b) = f (a + b)$ for all $a, b \in G$ .

Examples

For any group $G$ , there is a group homomorphism $1_{G} : G \to G$ , given by $1_{G} (g) = g$ for all $g \in G$ . This homomorphism is always bijective.

For any group $G$ , there is a (unique) group homomorphism into the group ${e}$ with one element and the only possible group operation $e * e = e$ . This homomorphism is given by $g \mapsto e$ for all $g \in G$ . This homomorphism is usually not injective: it is injective if and only if $G$ is the group with one element. (Uniqueness is guaranteed because there is only one function, let alone group homomorphism, from any set $X$ to a set with one element.)

For any group $G$ , there is a (unique) group homomorphism from the group with one element into $G$ , given by $e \mapsto e_{G}$ , the identity of $G$ . This homomorphism is usually not surjective: it is surjective if and only if $G$ is the group with one element. (Uniqueness is guaranteed this time by the property proved below that the identity gets mapped to the identity.)

For any group $(G, +)$ , there is a bijective group homomorphism to another group $G^{o p}$ given by taking inverses: $g \mapsto g^{- 1}$ . The group $G^{o p}$ is defined to have underlying set equal to that of $G$ , and group operation $g +_{o p} h := h + g$ .

For any pair of groups $G, H$ , there is a homomorphism between $G$ and $H$ given by $g \mapsto e_{H}$ .

There is only one homomorphism between the group $C_{2} = {e_{C_{2}}, g}$ with two elements and the group $C_{3} = {e_{C_{3}}, h, h^{2}}$ with three elements; it is given by $e_{C_{2}} \mapsto e_{C_{3}}, g \mapsto e_{C_{3}}$ . For example, the function $f : C_{2} \to C_{3}$ given by $e_{C_{2}} \mapsto e_{C_{3}}, g \mapsto h$ is not a group homomorphism, because if it were, then $e_{C_{3}} = f (e_{C_{2}}) = f (g g) = f (g) f (g) = h h = h^{2}$ , which is not true. (We have used that the identity gets mapped to the identity.)

Properties

The identity gets mapped to the identity. (Proof.)

The inverse of the image is the image of the inverse. (Proof.)

The image of a group under a homomorphism is another group. (Proof.)

The composition of two homomorphisms is a homomorphism. (Proof.)