I'm trying to find the number of automorphisms of Z8. When I google around, I find stuff like:
There are 4 since 1 can be carried into any of the 4 generators.
The problem hint tells me to make use of the fact that, if G is a cyclic group with generator a and f: G-->G' is an isomorphism, we know that f(x) is completely determined by f(a).
Thing is, I can think of 7! 1-1 and onto mappings of Z8 onto itself. I guess I don't see exactly why 1 has to get carried into a generator...why can't I have f(n) = n + 1 (mod 8), just shifting each element one to the right?
Thanks for any guidance on this, Mariogs