[Math] Prove that a cyclic group can have no more than one element of order two.

Question

(1)Why can't a cyclic group have more than one element of order two?

(2)Why does the group $U(n^2 -1)$ have to have more than one element of order two?

Answer 1

A cyclic group is addition modulo the number of elements. That number has at most one integer that is one half it.