# A bijection between automorphisms of a cyclic group and a multiplicative group

This is one more homework question I have--just another one I'm having trouble getting started with.

I'm supposed to prove a bijection between the multiplicative group of integers mod $p$ and the group of automorphisms of the cyclic group of order $p$.

Just don't know where to start--presumably something involving prime factorization, but honestly I'm just struggling with the intuition right now.

-

Let $\varphi:\mathbb{Z}_p \to \mathbb{Z}_p$ be an endomorphism (homomorphism with domain=codomain).

Consider $\varphi(1)=m$. Then $\varphi(1+1)=\varphi(1)+\varphi(1)=m+m=2m$ etc. So $\varphi(k)=km$.

So all homomorphisms are of the form $x \mapsto mx$ for some $m$. Call such a map $\varphi_m$.

When is $\varphi_m$ an automorphism? (What restriction must be placed on $m$?)

Edit: Oops! I didn't notice you're only looking to prove there is a bijection. The question:

What is $\varphi_m \circ \varphi_n$?

is to lead you to conclude that the bijection is actually an isomorphism.

-
Haven't got it yet but I think this gives me something good to think about on my walk home--although it might be the case that I need more help later, I think I can see a path from here. –  flury Nov 8 '11 at 0:58