Definition:Group Homomorphism

Definition
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.

Also defined as
Many sources demand further that $\map \phi {e_G} = e_H$ 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