I was just playing with the automorphism groups of cyclic groups, and I'm finding that perhaps they're not as straightforward as I thought they might be. Does anyone know if there's a formula to determine the automorphism group of in general?

For the prime case, how is this 'conjecture' of mine?

, prime

Also...I'd like to put a little plug in for my thread on automorphisms of subfields, as I'd really like an answer and it seems to have fallen out of interest...