Find Aut(Z). I want to find Aut(Z) so I listed the generators of Z which are 1, and -1. Then from the hint in the book it says once I know where the generators are mapped then I can get where every other element gets mapped. So I said the only choices for the automorphisms are defined by a(1) = 1 or a(1) = -1. How do I know that these are the only automorphisms? If I could say that an automorphism maps a generator to a generator that would help, but I dont really know that and cant prove it right now. Any thoughts?