# If $G = \mathbb{Z}_{p}^{*}$, show that $G$ is cyclic

On the group of the invertible elements in $$\mathbb{Z}_p$$, the question asks to show that the group is cyclic. This must have something to do with the representation of $$G$$ as a product of groups with prime power order, and I think that I should be able to find an element that generates the group, but so far no idea on how to do that.

G is abelian, by the way.

• There are lots of proofs of this. Many depend on Euler's $\varphi$ function that counts the number of integers less than $n$ and relatively prime to $n$. You can use it to count the number of generators of the cyclic subgroup generated by an element of $\mathbb{Z}_p^*$. That's the start of one proof. – Ethan Bolker Feb 21 at 0:05
• – lhf Feb 21 at 0:08