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# aug31 - -θ i so θ j 6 = ± θ i Since | j | = 4 we see...

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Math 5125 Wednesday, August 31 August 31, Ungraded Homework Exercise 4.4.4 on page 137 Prove | Aut ( Q 8 ) | ≤ 24. We write Q 8 in the usual way as 1 , ± i , ± j , ± k } . Since | i | = 4 and j / i , we see that Q 8 = i , j . Let θ be an automorphism of Q 8 . Since | i | = 4 and there are six elements of order 4, we see that there are at most six possibilities for θ i . Note that θ ( - i ) =
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Unformatted text preview: -θ i , so θ j 6 = ± θ i . Since | j | = 4, we see that θ j also has order 4 and consequently there are at most four possibilities for θ j . Since θ is determined by its affect on { i , j } , we conclude there are at most 6*4 = 24 possibilities for θ and the result follows....
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