There is a short proof that in GF(pq) the function φ(x) = xp is an automorphism. See this. I want to understand the entire group of automorphisms. As a permutation, φ has order q, which is to say that xpq = x for all x.
If ψ is an automorphism and we know ψ(X) then ψ(x) is determined for all other x.
As you can see this notation is very confusing. I shall attempt to see if lambda notation works. I recommence:
There is a short proof that in GF(pq) φ = λx.xp is an automorphism. Within the group of automorphisms, φ has order q for φq is the identity; i.e. xpq = x for all x.
If ψ is an automorphism then ψ(λx.x) entirely determines ψ. Given ψ(λx.x), ψ(y) = ξ