In English: Various Mappings in Abstract Algebra
Definitions
Let’s talk about a group $\left< G , \ast\ \right> , \left< G' , *' \right>$ and refer to it as $\phi : G \to G'$.
- If $\forall x ,y \in G $, $\phi (x \ast\ y) = \phi (x ) *' \phi ( y)$ then we call $\phi$ a Homomorphism.
- If a homomorphism $\phi$ is injective, then we call $\phi$ a Monomorphism and denote it $G \hookrightarrow G'$.
- If a homomorphism $\phi$ is surjective, then we call $\phi$ an Epimorphism and denote it $G \twoheadrightarrow G'$.
- If a homomorphism $\phi$ is bijective, then we call $\phi$ an Isomorphism and denote it $G \simeq G'$.
- For a homomorphism $\phi$, if $G = G'$ then we call $\phi$ an Endomorphism.
- For an isomorphism $\phi$, if $G = G'$ then we call $\phi$ an Automorphism.
Description
Although you might get overwhelmed by these sudden definitions, you’ll get used to them soon. Don’t be intimidated and face them confidently.
Monomorphisms and epimorphisms are arbitrarily termed, and in Japanese mathematics, they simply use モノ射 and エピ射 respectively. Outside of abstract algebra, these terms are basically used as injective and surjective functions, but in abstract algebra, they usually include homomorphisms.
The isomorphism is immediately useful for its properties, although the conditions required for it are a drawback. It would be better if those conditions could be reduced, meaning if it were enough to just have a monomorphism or an epimorphism.