Question

Prove that a group has exactly one element with the property that

x * x = x (this is called an idempotent element)

Answer 1

Let $x*x=x$ for any element $x$ of a group $G.$ Let $e$ be an identity element of the group $G$. Let $x\in G$ be arbitrary. It follows that $e*x=x$. On the other hand, $x*x=x$. Therefore, $x*x=e*x.$ The right cancellation property of a group implies that $x=e$ for all $x\in G$. Consequently, $G$ has exactly one element.