Various Maps in Abstract Algebra
Definition
For groups $\left< G , \ast\ \right> , \left< G' , *' \right>$, let $\phi : G \to G'$.
- If $\forall x ,y \in G $, $\phi (x \ast\ y) = \phi (x ) *' \phi ( y)$, then $\phi$ is called a homomorphism.
- If a homomorphism $\phi$ is injective, then $\phi$ is called a monomorphism and is written $G \hookrightarrow G'$.
- If a homomorphism $\phi$ is surjective, then $\phi$ is called an epimorphism and is written $G \twoheadrightarrow G'$.
- If a homomorphism $\phi$ is bijective, then $\phi$ is called an isomorphism and is written $G \simeq G'$.
- For a homomorphism $\phi$, if $G = G'$, then $\phi$ is called an endomorphism.
- For an isomorphism $\phi$, if $G = G'$, then $\phi$ is called an automorphism.
Explanation
Your head may ache from the flood of definitions, but you’ll get used to them soon, so don’t overthink it and face them head-on with confidence.
Monomorphism and epimorphism are translated arbitrarily here; even in the Japanese mathematical community they are simply written as モノ射 or エピ射. Outside of abstract algebra these terms themselves are used to mean injection and surjection respectively, but in abstract algebra a homomorphism is usually included.
The isomorphism has the drawback that its conditions are demanding, as much as its properties are immediately useful. In studying theory, it would be better if we could relax such conditions, that is, if a monomorphism or an epimorphism were sufficient.