Proof of Fermat's Little Theorem: g Generator of Fp*

[jacquelinek]

Prove that:
g is a generator of Fp* if and only if g^(p-1) = 1 (mod p) and gq ≠ 1 (mod p) for all prime divisors q of (p – 1).

I am thinking about applying Fermat's theorem...but don't know how...
Request help, thanks.

 

[Hurkyl]

Should that be g^q?

Fermat's theorem may be useful for part of the proof. It certainly cannot prove this theorem all by itself.

What do you know about finite fields? And about their unit groups?