Is there a way to apply the orbit-stabilizer formula to conclude that for $H,K \leq G$, then $[G: H\cap K]\leq [G:H][G:K]$?
The inequality isn't too hard to see, just by taking the map from $g/(H\cap K)\to G/H\times G/K$ defined as $ g(H\cap K)\mapsto (gH,gK) $ which is well defined and injective. I want to know if there is perchance a kind of combinatorial argument using the orbits and stabilizers under a group action of some sort. Cheers.