Kernel is Normal Subgroup of Domain

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Theorem

Then the kernel of $\phi$ is a normal subgroup of the domain of $\phi$:

$\ker \left({\phi}\right) \triangleleft \operatorname{Dom} \left({\phi}\right)$

Let $\phi: G_1 \to G_2$ be a group homomorphism, where the identities of $G_1$ and $G_2$ are $e_{G_1}$ and $e_{G_2}$ respectively.

By Kernel is Subgroup, $\ker \left({\phi}\right) \le \operatorname{Dom} \left({\phi}\right)$.

Let $k \in \ker \left({\phi}\right), x \in G_1$. Then:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \phi \left({x k x^{-1} }\right)$	$=$	$\displaystyle \phi \left({x}\right) \phi \left({k}\right) \left({\phi \left({x}\right)}\right)^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Homomorphism to Group Preserves Inverses
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \phi \left({x}\right) \left({\phi \left({x}\right)}\right)^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	as $k \in \ker \left({\phi}\right)$
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \phi \left({x}\right) \left({\phi \left({x}\right)}\right)^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Property of Identity
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle e_{G_2}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Property of Inverse

So $x k x^{-1} \in \ker \left({\phi}\right)$.

As this is true for all $x \in G_1$, then from Normal Subgroup Equivalent Definitions, $\ker \left({\phi}\right)$ is a normal subgroup of $G_1$.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \phi \left({x k x^{-1} }\right)\)	\(=\)	\(\displaystyle \phi \left({x}\right) \phi \left({k}\right) \left({\phi \left({x}\right)}\right)^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Homomorphism to Group Preserves Inverses
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \phi \left({x}\right) \left({\phi \left({x}\right)}\right)^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	as $k \in \ker \left({\phi}\right)$
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \phi \left({x}\right) \left({\phi \left({x}\right)}\right)^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Property of Identity
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle e_{G_2}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Property of Inverse