Full disclosure: this is a homework problem, but it is not assigned to turn in for credit. The problem is from Dummit and Foote, Chapter 3.2:
Suppose $H, K$ are subgroups of finite index a group (not necessarily finite) group $G$ with $[G:H] = m$, $[G:K] = n$. Prove that LCM($m,n$) $\le [G:H\cap K] \le mn $.
I looked at this post, but I'm still somewhat confused. I don't understand how to apply the Orbit-Stabilizer Theorem.
Letting $G$ act on $G/H \times G/K$ by left multiplication, the stabilizer of $(H,K)$ is clearly $(H\cap K)$, I see that much. Thus, $[G:H \cap K] = |Orb(H,K)|$, but how do I determine the length of the orbit?
After a very helpful hint from Mohan I saw what I was missing and the answer is rather simple.
For the second inequality, $Orb(H,K) \subset G/H \times G/K$, so $|Orb(H,K)| \le mn$.
For the first inequality, since:
$$ [G:H \cap K] = [G:H][H:H \cap K] = [G:K][K:K \cap H] $$
So $m,n$ both divide $[G:H \cap K]$.