Definition:Group Homomorphism

Definition
Let $\left({G, \circ}\right)$ and $\left({H, *}\right)$ be groups.

Let $\phi: G \to H$ be a mapping such that $\circ$ has the morphism property under $\phi$.

That is, $\forall a, b \in G$:
 * $\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$

Then $\phi: \left({G, \circ}\right) \to \left({H, *}\right)$ is a group homomorphism.

Also defined as
Many sources demand further that $\phi \left({e_G}\right) = e_H$ as well, where $e_G$ and $e_H$ are the identity elements of $G$ and $H$, respectively.

However, this condition is superfluous, as shown on Group Homomorphism Preserves Identity.

Also known as
Some sources refer to a homomorphism as a representation.

Some sources use the term structure-preserving.

Also see

 * Definition:Homomorphism (Abstract Algebra)
 * Definition:Ring Homomorphism


 * Definition:Group Epimorphism: a surjective group homomorphism


 * Definition:Group Monomorphism: an injective group homomorphism


 * Definition:Group Isomorphism: a bijective group homomorphism


 * Definition:Group Endomorphism: a group homomorphism from a group to itself


 * Definition:Group Automorphism: a group isomorphism from a group to itself