# Automorphism Group of Paley Graph

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

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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 $-b$.) – user1729 Jul 25 '11 at 14:48
(I am assuming $p=1 \text{ mod } 4$, as otherwise $b-a$ is not a square...) – user1729 Jul 25 '11 at 14:55
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. – Lost_DM Jul 25 '11 at 15:15
Crossposted to MO: mathoverflow.net/questions/71236/… – Zev Chonoles Jul 25 '11 at 15:36