Kernel

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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 $G'$.

**Example(s)**

Let $\phi: \mathbb{Z} \longrightarrow \mathbb{Z}_n$. $m \longrightarrow m\ mod\ n$ is a homomorphism.

$Ker(\phi) = \{kn\ |\ k\in \mathbb{Z}\}=n\mathbb{Z}=\langle n \rangle$.

**Non-example(s)**

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**Additional Comments**

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