\begin{definition}[Definition:Group Homomorphism]
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: G \to H$ be a mapping such that $\circ$ has the morphism property under $\phi$.
That is, $\forall a, b \in G$:
:$\map \phi {a \circ b} = \map \phi a * \map \phi b$
Then $\phi: \struct {G, \circ} \to \struct {H, *}$ is a group homomorphism.
\end{definition}