I'm working on proving the following claim:
"Let $r \in U(n)$ and $\forall s \in \mathbb{Z_n}$, $\alpha: \mathbb{Z_n} \rightarrow \mathbb{Z_n}$ defined by $\alpha(s)=rs\mod{n}$ is an automorphism."
I've already proven one-to-one and onto, but can't quite figure out how to show that the mapping preserves operations (it intuitively seems obvious, I think). Anyway, here's what I've got:
$\alpha(a+b)=[r(a+b)]\mod{n}=(ra+rb)\mod{n}=(ra\mod{n}+rb\mod{n})\mod{n}=(\alpha(a) + \alpha (b))\mod{n}.$
But, $ (\alpha(a) + \alpha (b))\mod{n}$ doesn't necessarily equal $\alpha(a)+\alpha(b)$. Consider $\alpha(a)=n-1$ and $\alpha(b)=1$, then $\alpha(a) + \alpha (b)=n \not \in \mathbb{Z_n}$ .
Am I going about this wrong? How do I correctly show that $\alpha$ preserves the operation?
But, The question is to show that the map $\alpha$ is well defined, i.e if $s=s' $ mod $n$ then $rs=rs'$ mod $n$.