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!

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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 $m$.