Let G be a finite group and =G X G. There is an action of G on such that g.(a,b)=(ga,gb) for gG and (a,b) .

Definition of Orbits - Let G be a group and let be a set. Assume that G acts on and let G. The orbit of in this action is denoted by O() or G and it is defined as

O() = G = { | there exists gG with g. = } The set of orbits of the action of G on is denoted by G

Show that the number of orbits for this action equals |G|.