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Formal Definition

Let $\phi\ :\ G\longrightarrow G'$ be a homomorphism groups. The subgroup $\phi^{-1}[\{e'\}]=\{x\in G|\phi (x) = e'\}$ is the kernel of $\phi$, denoted by $Ker(\phi)$.

Informal Definition

$Ker(\phi)$ is the subset of $G$ that maps the identity of $G'$.


Let $\phi: \mathbb{Z} \longrightarrow \mathbb{Z}_n$. $m \longrightarrow m\ mod\ n$ is a homomorphism.
$Ker(\phi) = \{kn\ |\ k\in \mathbb{Z}\}=n\mathbb{Z}=\langle n \rangle$.


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