Subgroup of Abelian Group is Normal

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Every subgroup of an abelian group is normal.


Let $G$ be an abelian group.

Let $H \le G$ be a subgroup of $G$.

Then for all $a \in G$:

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