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 $H\cup K\subseteq HK\cap KH$.)