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

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# Definition of Homomorphism

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

${\displaystyle f\colon G\to K}$ is homomorphism iff

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

# Definition of Kernel

Let eK be identity of K

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

# Definition of Image

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