So I'm asked to find the number of automorphisms (homomorphism mapping it to itself) in the cyclic group under addition.

I think this should be 4, to correspont to <1>,<3>,<5>, and <7>, because these are relatively prime to 8 and act as generators for the group, and I know that the homomorphism should map generators to generators.

But I'm kind of unsure as to why this is. I don't know how to prove it/can't find more information about it. Am I right? Any insight?