I'm starting doing my Group theory's course problems, and I'm having trouble with this one:
Consider the multiplicative group $\mathbb{Z}^{\times}_n$ of the units of the ring $\mathbb{Z}_n$. Prove that $\mathbb{Z}^{\times}_n$ is cyclic if n is prime.
The notation used means $\mathbb{Z}^{\times}_n=\{r\in\mathbb{Z}_n:\exists s\in\mathbb{Z}_n \text{ s.t } rs=1=sr\}$. I guess it will have somewhat to do with Euler's $\phi$ function. I first suppose $n$ prime, but know what's the next step. Any help will be appreciated.