2-Group with Exponent 2 is Abelian

Proposition

Let $G$ be a $2$-group. If $\exp (G)=2$, then $G$ is abelian.

\begin{proof} For any $1\ne x,y\in G$, since $x=x^{-1}$ and $y=y^{-1}$, we have that

$$[x,y]=xy{x}^{-1}{y}^{-1}=(xy)(xy)=1$$

and so $G$ is abelian. \end{proof}