I came across one problem in GTM244 about the circulant graph:
A circulant is a Cayley graph $CG(\mathbb{Z}_n,S)$, where $\mathbb{Z}_n$ is the additive group of integers modulo $n$. Let $p$ be a prime, and let $i$ and $j$ be two nonzero elements of $\mathbb{Z}_p$.Determine when $CG(\mathbb{Z}_p,\{1,-1,i,-i\})\cong CG(\mathbb{Z}_p,\{1,-1,j,-j\})$
I only can show when $i\equiv\pm j$ or $ij\equiv\pm 1$, they are isomorphic but how can I find all possibilities? Intuitively, the $p$-cycle $0-1-2-\cdots-p$ and $0-i-2i-\cdots-pi$ should be mapped to $0-1-2-\cdots-p$ and $0-j-2j-\cdots-pj$ under an isomorphism.Is that the right idea? How to prove it? Thanks for any help.