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 with only one generator can have at most 2 elements. Let $a$ be a generator of a cyclic group. Since the order of $a^{-1}$is equal to the order of $a,$ we conclude that $a^{-1}$is also a generator, and hence $a^{-1}=a.$ It follows that $a^2=e$ after multiplying both parts by $a.$ Consequently, a cyclic group can have at most 2 elements.