Subgroup Containing all Squares of Group Elements is Normal

Theorem
Let $G$ be a group.

Let $H$ be a subgroup of $G$ with the property that:
 * $\forall x \in G: x^2 \in H$

Then $H$ is normal in $G$.

Proof
We have:

Because $\paren {x h}^2$ and $\paren {x^{-1} }^2$ are in the form $x^2$ for $x \in G$, they are both elements of $H$.

Thus:
 * $x h x^{-1} \in H$

and so $H$ is normal in $G$ by definition.