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 $Q=\left\{\mathbf 1,-\mathbf 1,\mathbf i,-\mathbf i,\mathbf j,-\mathbf j,\mathbf k,-\mathbf k\right\}$ has only one element of order two, which is $-\mathbf 1$, 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.

$\blacksquare$

Sources

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 46 \lambda$