Question

Prove that a cyclic group with only one generator can have at most 2 elements.

Answer 1

Let us prove that a cyclic group $G$ with only one generator can have at most 2 elements. If $G$ is infinite cyclic group, then $G$ is isomorphic to the group $(\mathbb Z,+)=\langle1\rangle=\langle -1\rangle$, and hence it has two generators. Therefore, $G$ must be a finite cyclic group. Let $a$ be a generator of a cyclic group , and let $|a|=n.$ Taking into account that $|a^{-1}|=n$ and $a$ is a unique generator of $G,$ we conclude that $a^{-1}=a.$ It follows that $a^2=e,$ and hence $G$ is a cyclic group of order 2, or $G$ is a trivial group. We conclude that $G$ can have at most 2 elements.