Abstract Algebra/Group Theory/Homomorphism/Kernel of a Homomorphism is a Normal Subgroup

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Theorem[edit]

Let f be a homomorphism from group G to group K. Let eK be identity of K.


 \text{ker} ~f is a normal subgroup.

Proof[edit]

f(g \ast n \ast g^{-1}) = f(g) \circledast f(n) \circledast f(g^{-1}) = f(g) \circledast e_K \circledast f(g^{-1}) = f(g) \circledast f(g^{-1}) = f(g \ast g^{-1}) = f(e_{G}) = e_K