I can not solve the following two problems on Group Theory.
A subgroup $H$ of a group $G$ has the property that $x^2 \in H$ for every $x\in G$. Prove that $H$ is normal in $G$ and $G/H$ is abelian.
Let $G$ be a group of order 8 and $x$ be an element of $G$. Prove that $x^2$ is in the center of $G$.