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

Definition

Formally, given two groups \((G, +)\) 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 \(G^{\mathrm{op}}\) given by taking inverses: \(g \mapsto g^{-1}\). The group \(G^{\mathrm{op}}\) is defined to have underlying set equal to that of \(G\), and group operation \(g +_{\mathrm{op}} h := h + g\).

• 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 \(e_{C_3} = f(e_{C_2}) = f(gg) = f(g) f(g) = h h = h^2\), 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.)