Let G be a finite group
a) Show that the number of elements x of G such that x^3=e is odd
Let . Now if is in then so has order . Therefore, . But . Thus, . Let . Write them in such a way so that . Therefore, all can be paired with . Thus, there are an even number (zero is allowed) of elements which have order three. But we also include and this makes the total number of elements odd.
b) Show that the number of elements x of G such that x^2 does not equal e is even