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I would like an explanation as to the structure description of the automorphism group of a Paley graph.

Paley graphs are a specific case of Cayley graphs where the group is $Z_q$ (q is a prime power for some prime p = 1 mod 4) and the connection set is all the quadratic residues in GF(q).

I'll be satisfied even with the less general case where q is prime.

I'm pretty sure that the said group is a semi-direct product of CyclicGroup(q) and CyclicGroup(q-1/2) but I have trouble showing it in the general case...

Thanks!

P.S

Also posted on: https://mathoverflow.net/questions/71236/automorphism-group-of-paley-graph

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    For the prime case, is it not just $D_p$? (If $a-b=c^2$ then $a+1-(b+1)=a-b=c^2$, so rotation works, while flipping works because the flip takes $a$ to $-a$ and $b$ to $-b$.)2011-07-25
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    (I am assuming $p=1 \text{ mod } 4$, as otherwise $b-a$ is not a square...)2011-07-25
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    You are correct about $p=1 mod 4$. What you stated (correctly) is that $D_p$ is a subgroup (which indeed it is). However, it is not the whole group, and if I am correct it is a subgroup of index 4.2011-07-25
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    Crossposted to MO: http://mathoverflow.net/questions/71236/automorphism-group-of-paley-graph2011-07-25

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