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

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Definition of Homomorphism[edit]

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[edit]

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[edit]

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