Let G be a finite group
a) Show that the number of elements x of G such that x^3=e is odd
b) Show that the number of elements x of G such that x^2 does not equal e is even
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.
Similar to above.b) Show that the number of elements x of G such that x^2 does not equal e is even