order of finit group which has even elements

Question

I have a question,

prove that a finite group has an even number of elements, if and only if the group consists of an element of order $2$.

Answer 1

Hint: The identity element is the only element which has order 1. So assume $g \neq g^{-1}$ for all $g \neq e$ in the group. Then show that the size of the group is odd, since inverses are uniquely determined and the identity element is also in the group.