Anyone who can help?

Question

If * is a binary operation on a set S, an element x of S is an idempotent for * if x*x=x. Prove that a group has exactly one idempotent.

Answer 1

Let G be our group, e its identity and x an element of G.
Then e is an idempotent since e*e=e
Suppose x*x=x
Multiply both sides on the left by x^-1
Then x^-1x*x = x^-1*x = e
which implies
x = e.

Answer 2

As far as I can remember, it is a proof by contradiction. Suppose not . . . .

Oh well, maybe it's a start. ;)