Group Homomorphism

A group homomorphism is a function $\phi : G \to H$ between groups that preserves the group operation: $\phi(a \cdot b) = \phi(a) \cdot \phi(b)$ for all $a, b \in G$. This extends the notion of homomorphism from monoids to groups.

A group homomorphism automatically preserves the identity ($\phi(e_G) = e_H$) and inverses ($\phi(g^{-1}) = \phi(g)^{-1}$). These facts follow from the group axioms and do not need to be checked separately.

The kernel of $\phi$ is $\ker(\phi) = \{g \in G \mid \phi(g) = e_H\}$, which is always a normal subgroup of $G$. The image of $\phi$ is $\operatorname{im}(\phi) = \{\phi(g) \mid g \in G\}$, which is always a subgroup of $H$. A homomorphism is injective if and only if its kernel is trivial.

In category theory, group homomorphisms are the morphisms in the category of groups.