Abstract Algebra/Group Theory/Homomorphism/Definition of Homomorphism, Kernel, and Image

Definition of Homomorphism

Let G, K be groups with binary operations $\ast$ and $\circledast$ respectively.

$f\colon G\to K$ is homomorphism iff

• $\forall \;g_{1},g_{2}\in G:f(g_{1}\ast g_{2})=f(g_{1})\circledast f(g_{2})$

Definition of Kernel

Let e_K be identity of K

${\text{kernel}}~f={\text{ker}}~f=\lbrace g\in G\;|\;f(g)={\color {OliveGreen}e_{K}}\rbrace$

Definition of Image

${\text{Image}}~f={\text{im}}~f=\lbrace k\in K\;|\;\exists \;g\in G:f(g)=k\rbrace$