What can we say about the size of $HK\cap KH$ when $HK\neq KH$?

If $G$ is a finite group, and $H$, $K$ are proper subgroups of $G$, then it is not necessary that $HK=KH$. But, these two subsets have same size. The question I would like to ask, then, is

If $HK\neq KH$, then what can we say about the size of $HK\cap KH$?

(Note that $H\cup K\subseteq HK\cap KH$.)

Solution 1:

It would useful to connect $M=HK\cap KH$ with $N=H\cap K$. Evidently, $M$ is a (disjoint) union of double cosets $NxN, x\in M$. By [M.Hall, The Theory of Groups, Theorem 1.7.1] $$ |NxN|=\frac{|N|^2}{|N\cap x^{-1}Nx|}. $$ So, for example, if $p||N|$ ($p$ is prime) then $p||M|$.