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, $\forall a, b \in R$:

$\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$

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

Also see

• Homomorphism
• Ring Homomorphism

• Group Epimorphism: a surjective group homomorphism

• Group Monomorphism: an injective group homomorphism

• Group Isomorphism: a bijective group homomorphism

• Group Endomorphism: a group homomorphism from a group to itself

• Group Automorphism: a group isomorphism from a group to itself

• Results about group homomorphisms can be found here.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 7.1$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.10$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 60$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 47$
• John F. Humphreys: A Course in Group Theory (1996): $\S 8$: Definition $8.1$