Subgroup of Abelian Group is Normal

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Theorem

Every subgroup of an abelian group is normal.


Proof

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

Then:

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

$\blacksquare$


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