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.

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Comments

  • Hafsa Anwer  On January 26, 2012 at 8:44 am

    very good answer

  • Hafsa Anwer  On January 26, 2012 at 8:45 am

    it solved my problem.

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