Automorphism Groups: Finite Cyclic G of Order n

[camelite]

Homework Statement

If G is a finite cyclic group of order n, what is the group Aut(G)? Aut(Aut(G))?

Homework Equations

The Attempt at a Solution

Aut(G) is given by the automorphisms that send a generator to a power k < n where (k,n) = 1 with order p(n) where p is Euler's function.

I'm having trouble visualizing or describing Aut(Aut(G)) as automorphisms of automorphisms. Is Aut(G) isomorphic to a cyclic group of order p(n)?

 

[matt grime]

It is not going to be cyclic in general. You need to try to get a better view of Aut(G).

You've already identified it as a set as

{ k : 1<=k <n and (k,n)=1}

What is the group law?