# Cyclic group generators

My question is: Can you find a cyclic group with n generators?

I know that zero (or any other identity element for that matter) is included, so there would be for $Z_n$ at most n-1 generators. However, is it possible to say that $Z_{n+1}$ could provide n generators?

Any help would be much appreciated! If you wouldn’t mind, I would appreciate it if you provide enough detail to really help me understand so I can learn. Thanks in advance!

#### Solutions Collecting From Web of "Cyclic group generators"

For the trivial group, and the group with two elements, there is one generator. So the answer to your question is yes if $n=1$.

For cyclic groups of order larger than $2$, if $x$ is a generator, then $x^{-1}$ is also a generator, and $x\ne x^{-1}$ else $x$ has order $1$ or $2$ (contradiction). Hence, in this case, generators come in pairs and hence there must be an even number of them. So the answer to your question is no if $n>1$ and $n$ is odd.

If $n$ is even, then as the comments note, a cyclic group of order $m$ has $\varphi(m)$ generators, so we need to find some $m$ such that $\varphi(m)=n$. However, it is not always the case that such $m$ exists. Even $n$ that are not equal to $\varphi(m)$ for some $m$ are called “non-totients” and are listed in this sequence. So finally, the answer to your question is no if $n$ is even and a non-totient, and yes if $n=\varphi(m)$ for some $m$.