Prove that Cyclic Group with only one generator can have at most 2 element

Question

I want someone to check my proof Suppose the cyclic group with one generator can have more than 2 element Consider 2 element since number of generator can't exceed number of element then it has 2 element. But if it not,It must have <1> as generator and atleast

prime number as generator.That contradict Therefore cyclic group with only one generator can have at most 2 element

Answer 1

Take $G$ such a Group with its unique generator $g\in G$. But then $g^n=e\iff \left(g^{-1}\right)^n=e$ thus $g^{-1}$ is also a generator. Can you take from here?