# Can a cyclic group have more than two generators? [duplicate]

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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?

## marked as duplicate by Dietrich Burde, Community♦Apr 9 '15 at 14:09

• A cyclic, or not, group can have lots of sets of generators with different cardinalities, yet a cyclic group is characterized for having a generator set with one single element. Of course, a generator for a cyclic group and any other element(s) will also be a generator set. – Timbuc Apr 9 '15 at 13:59
• @435145 See the Wikipedia article on Cyclic groups! - en.wikipedia.org/wiki/Cyclic_group#Integer_and_modular_addition – Jesse P Francis Apr 9 '15 at 14:00

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.
• @435145: Ok, by writing expilictly, you can see that $Z_5$ has $4$ generator. i.e every nonidentity element is a generator. – mesel Apr 9 '15 at 13:59
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 $\phi(n)$.