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

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Prove that under any automomorpism of , r has at most two possible images and s has at most 4 possible images. Deduce that |Aut( )| 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 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 radian. Let s be the reflection about the line of symmetry through vertex 1 and the origin.

Then

= {1, r, , , s, sr, s , s }

Peter