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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.

Shaun
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dren's theorem
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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jan 28 '23 at 15:59
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2 Answers2

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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.

Igor Rivin
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    This question seems not to meet the standards for the site. Instead of answering it, it would be better to look for a good duplicate target, or help the user by posting comments suggesting improvements. Please also read the meta announcement regarding quality standards. – Shaun Jan 28 '23 at 15:39
  • @Shaun Why are you telling me this? – Igor Rivin Jan 28 '23 at 17:50
  • @Shaun You should know by now that I know at least as much as you do, and so refrain from lecturing. – Igor Rivin Jan 28 '23 at 18:58
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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 $S_4.$

mathlander
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  • This question seems not to meet the standards for the site. Instead of answering it, it would be better to look for a good duplicate target, or help the user by posting comments suggesting improvements. Please also read the meta announcement regarding quality standards. – Shaun Jan 28 '23 at 18:02