Determine the Automorphism group of the quaternion group.

Question

Question: Determine ${\rm Aut}(Q_{8})$ for $Q_{8}$ being the quaternion group. I'm currently self studying some abstract algebra and I am struggling a little bit with this question from Micheal Artin's Algebra book.

Answer 1

Well, 1 has to go to 1, -1 has to go to -1, since it is the only element of order two. Now see where $i, j, k$ can go.

Answer 2

The elements $1$ and $-1$ are fixed. We can imagine the remaining elements as an octahedron in $3$d space, where multiplying two opposite vertices gives $1$, multiplying a vertex by itself gives $-1$, and multiplying two adjacent vertices gives their cross product, which is basically finding the vertex such that drawing a loop for the first element to the second element to the product goes counterclockwise around a face, if we are looking from outside the octahedron. Therefore, a symmetry of the quaternion group corresponds to a rotational symmetry of the octahedron. Since the rotational symmetry group of an octahedron is $S_4,$ the automorphism group of $Q_8$ is $S_4.$