Help with classification of the set of automorphisms on Z/n

[TopCat]

Let G be a group and Aut G the set of all automorphisms of G. What is Aut Z_{n} for arbitrary n\inN?

Sol

Let f\inAut Z_{n}. Since Z_{n}=<1>, f can be completely characterized by f(1), i.e., f(k) = f(1)^{m}, for k\inZ_{n} and m\inN. By these facts, it follows that f(1) must generate Z_{n}, that is, Z_{n}=<f(1)>.

Define H as the group of all generators of Z_{n} under multiplication. Then f(1)\inH.

Now at this point I'm certain that Aut Z_{n}\congH, but I just can't find a homomorphism from Aut Z_{n}\rightarrowH. I can't concoct an epimorphism in the opposite direction either. The closest I got is the below:

Also define the map c:Aut Z_{n}\rightarrowH by c(f) = f(1). Choose f,g\inAut Z_{n}. Then:
c(g\circf) = g\circf(1) = g(f(1)) = g(1)^{l}. But any l'\equivl (mod n) \Rightarrowg(f(1)) = g(1)^{l&#039;}, so we can consider l' an equivalence class in Z_{n}. That implies l = f(1)^{p} for some p\inN, so that c(g\circf) = f(1)^{p}*g(1) = c(g)c(f)^{p}.

I can't see what I'm overlooking, so I'd appreciate if someone can help point me in the right direction. Thanks!

 

[Robert1986]

Well, H is just Z/nZ isn't it? Find an isomorphism from Aut(Z_n) to Z/nZ. You know that if if is in Aut(Z_n) then it is uniquely determined by what it does to 1. For example, f(1) = a*1 for some a in Z. Now try to use this to construct an isomorphism from Aut(Z_n) to Z/nZ.

 

[TopCat]

Not quite. H\subsetZ_{n}, but not a subgroup since Z_{n} is additive and H is multiplicative. In particular, H is always a proper subset since 0\inZ_{n} but 0\notinH. So I have to find an isomorphism from Aut Z_{n} directly to H.