Proving or disproving that a nonidentity cyclic group has at least two generators.

Question

I had trouble proving that a nonidentity cyclic group has at least two generators, but I am starting to think that it has to be disproven. Would I have to disprove it by showing that any cyclic group has at least two generators besides the identity?

Answer 1

[A] nonidentity cyclic group has at least two generators.

The statement is false. Consider the cyclic group of two elements, given by the presentation $$\langle a\mid a^2\rangle,$$ also known as $\Bbb Z/2\Bbb Z$ through isomorphisms (where $a\mapsto [1]_2$). This has exactly one generator.

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As pointed out in the comment by @Bernard, the statement is true of cyclic groups of order at least three. (See here.)

Answer 2

A cyclic group of order $n$ has $\varphi(n)$ generators, where $\varphi$ is Euler's totient function. For $n\ge3$, $\varphi(n)\ge2$.