Dummit and Foote Section 4.4 Automorphisms, Exercise 3 reads as follows:

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Prove that under any automomorpism of $\displaystyle D_8$, r has at most two possible images and s has at most 4 possible images. Deduce that |Aut($\displaystyle D_8$)| $\displaystyle \leq$8

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Can anyone help with the above problem? Be grateful for help!

The notation of D&F regarding the construction of $\displaystyle D_8$ is as follows:

Fix a square centred at the origin in an x,y plane and label the vertices from 1 to 4 in a clockweise manner. Let r be the rotation clockwise about the origin through $\displaystyle \pi /2$ radian. Let s be the reflection about the line of symmetry through vertex 1 and the origin.

Then

$\displaystyle D_8$ = {1, r, $\displaystyle r^2$, $\displaystyle r^3$, s, sr, s$\displaystyle r^2$, s$\displaystyle r^3$}

Peter