## Q(8) has at most 24 distinct automorphisms

Show that $|\mathsf{Aut}(Q_8)| \leq 24$.

Recall that $Q_8 = \langle i,j \rangle$, and let $\varphi$ be an automorphism of $Q_8$.

We have $|i| = |j| = 4$. Then $|\varphi(i)| = |\varphi(j)| = 4$. There are 6 elements of order 4 in $Q_8$, so we have 6 choices for $\varphi(i)$. Note that if we choose $\varphi(j) \in \langle \varphi(i) \rangle$, then $\varphi$ is not surjective, hence not an isomorphism. Two of the six elements of order 4 are in $\langle \varphi(i) \rangle$, so we have 4 choices for $\varphi(j)$. Now $\varphi$ is determined by its image at $i$ and $j$, so that $|\mathsf{Aut}(Q_8)| \leq 24$.