CAPS 1996 Problem 2

Let ⋆ be a binary operation on a nonempty set $M$ . That is, every pair $(a,b) \in M$ is assigned an element $a$ ⋆ $b$ in $M$ . Suppose that ⋆ has the additional property that $(a$ ⋆ $b)$ ⋆ $b= a$ and $a$ ⋆ $(a$ ⋆ $b)= b$ for all $a,b \in M$ . (a) Show that $a$ ⋆ $b = b$ ⋆ $a$ for all $a,b \in M$ . (b) On which finite sets $M$ does such a binary operation exist?