Quaternion Group not Dihedral Group

Theorem
Let $Q$ be the quaternion group.

Then $Q$ is not isomorphic to the dihedral group $D_4$.

Proof
From Generator of Dihedral Group we know that $D_4$ is generated by two elements of orders $4$ and $2$ respectively. Let's call them $\alpha$ and $\beta$ where $\alpha^4=e$ and $\beta^2=e$.

Hence $\alpha^2$ and $\beta$ are diferent elements of $D_4$ which both have order two.

But the quaternion group $Q$ has only one element of order two, which is $-1$, the rest have order one or four.

Thus $Q$ and $D_4$ can not be isomorphic because there are not the same number of elements of order two in both groups.