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

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