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This question already has an answer here:

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?

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

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  • $\begingroup$ 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. $\endgroup$ – Timbuc Apr 9 '15 at 13:59
  • $\begingroup$ @435145 See the Wikipedia article on Cyclic groups! - en.wikipedia.org/wiki/Cyclic_group#Integer_and_modular_addition $\endgroup$ – Jesse P Francis Apr 9 '15 at 14:00
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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.

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  • $\begingroup$ Sorry, I don't understand that, I have just a basic knowledge about group theory. $\endgroup$ – MrDi Apr 9 '15 at 13:58
  • $\begingroup$ @435145: Ok, by writing expilictly, you can see that $Z_5$ has $4$ generator. i.e every nonidentity element is a generator. $\endgroup$ – mesel Apr 9 '15 at 13:59
  • $\begingroup$ Maybe to be a bit clearer: Every finite cyclic group has a generator for each number coprime to the order of the group, and less than the order. Since 5 is prime, all numbers less than 5 are coprime, so there are 4 generators. On the other hand, the cyclic group of order 10 also has 4 generators: 1, 3, 7 and 9. $\endgroup$ – Alfred Yerger Apr 9 '15 at 14:04
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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 $\phi(n)$.

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