Subgroup of Abelian Group is Normal

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Theorem

Let $H \le G$ where $G$ is abelian.

Then:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle y$	$\in$	$\displaystyle H^a$	$\displaystyle $	$\displaystyle $	$\displaystyle $	where $H^a$ is the conjugate of $H$ by $a$
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle a y a^{-1}$	$\in$	$\displaystyle H$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Conjugate
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle y$	$\in$	$\displaystyle H$	$\displaystyle $	$\displaystyle $	$\displaystyle $	because $a y a^{-1} = y$ as $G$ is abelian

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle y\)	\(\in\)	\(\displaystyle H^a\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	where $H^a$ is the conjugate of $H$ by $a$
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle a y a^{-1}\)	\(\in\)	\(\displaystyle H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Conjugate
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle y\)	\(\in\)	\(\displaystyle H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	because $a y a^{-1} = y$ as $G$ is abelian