Can a cyclic group have more than two generators?

Question

Can a cyclic group have more than two generators?

for example the group $\mathrm{Z}$ has two generators $-1$ and $1$, but can a group have more than two generators?

Answer 1

Yes, take $G=\mathbb Z_5$ then it has a $\phi(5)=4$ generater.

In general $Z_n$ has $\phi(n)$ generater where $\phi$ is Euler phi function.

Answer 2

Sure. Recall that in $\mathbb{Z}/n$, the order of $a$ is $\frac{n}{\gcd(n,a)}$. Therefore the number of generators of $\mathbb{Z}/n$ is just the number of elements prime to $n$, which is $\phi(n)$.