TITLE: Powers as a complete residue system modulo $p$?
QUESTION [5 upvotes]: Question 1.
With $0 < a < p$, $p$ prime and $\gcd(a,p-1)=1$, is it true that $0, 1, 2^a, ...,(p-1)^a$  is a complete residue system modulo $p$?  If not, will a similar statement hold?
Question 2. 
I was told it works for $a = 3$, does anyone know a simple proof of it in this particular case?

REPLY [3 votes]: HINT $\rm\ (a,p-1)=1\ \Rightarrow\ \exists\: b,\ b\:a\:\equiv\: 1\pmod{p-1}\ \Rightarrow\ (n^b)^{\:a}\equiv\: n\pmod{p}\ $ so $\rm\:x\:\to\: x^{\:a}\:$ is onto