number of orbits of a group: is map well-defined and surjective?

[mona123]

Let $G$ be a finite group acting on a finite set $X$. Let $m$ be a number of orbits of $G$ on $X$ and $M$ be the number of orbits of $G$ on $X\times X$. Show that $m^2\le M$ with equality if and only if G acts trivialy on $X$.


I need your help to solve this problem. Thanks.

 

[mona123]

For that i defined the map $\mathcal{O}_{X\times X}\to \mathcal{O}_X\times \mathcal{O}_X$, $[(x,y)] \mapsto ([x],[y])$


where $O_Z$ denotes the set of orbits of $Z$ and $[z]$ denotes the orbit $\{g\cdot z : g\in G\}$


I want to check that this map is well-defined and surjective.
Please hepl me