# Group action on a Cartesian product

Let $G$ be a finite group acting on a finite set $X$. Then naturally $G$ acts on $X \times X$ by $g.(x,y)=(g.x,g.y)$. Is there any way to find the number of orbits of the action of $G$ on $X\times X$ using the action $g$ on $X$? Are they related?

• Not sure if this is the best available answer but it's a good start point groupprops.subwiki.org/wiki/Orbit-counting_theorem Using the fact that Stab$((s,t))$ ($G$ acting on $X\times X$) is Stab$(s)\cap$Stab$(t)$ ($G$ acting on $X$) gives an explicit formula. – Robert Chamberlain Sep 19 '17 at 19:07
• Using $|\mathrm{Fix}_{X\times X}(g)|=|\mathrm{Fix}_{X}(g)|^2$ also gives an explicit formula. – Robert Chamberlain Sep 19 '17 at 19:19
• This is not a general answer, but if the action of $G$ on $X$ is doubly transitive then you can show there are precisely 2 orbits – TomGrubb Sep 19 '17 at 19:19