# Subgroup of Abelian Group is Normal

## 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

$\blacksquare$