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

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