The cyclic group $(\Bbb{Z}_n, +)$ of order $n\ge2$ has exactly one generator if and only if the group is $(\Bbb{Z}_2, +)$.

Question

For a cyclic group $(\Bbb{Z}_n, +)$ of order $n\ge2$, there exists an element $g$ of $(\Bbb{Z}_n, +)$ such that $\{g^n: n\text{ is an element of }\Bbb{Z}\} = (\Bbb{Z}_n, +)$. Such an element $g$ is called a generator of the cyclic group.

Answer 1

Let $G$ be cyclic group of order $n\in\mathbb N$, $n>2$. Let be $g\in G$ a generator. Then $g^{-1}$ is a generator, too, and $g \neq g^{-1}$ because otherwise $g$ would be of order $2$.