I asked this question two days ago om Math SE but didn't receive an answer: https://math.stackexchange.com/questions/1597321/generalization-of-hkh-kk-cap-h

Suppose we are given subgroups $H,K$ of a finite group $G$. Denote by $\langle H,K\rangle$ the subgroup generated by $H$ and $K$.

If $H$ normalizes $K$, then $\langle H,K\rangle =HK$ and $$(\langle H,K\rangle:H) = (K:H \cap K)\tag{$\ast$}$$ Is there a generalization of $(\ast)$ if $H$ isn't supposed to normalize $K$ ?

Comment: $(\ast)$ always holds for $HK$ on the left hand side (in place of $\langle H,K\rangle$) by he double coset formula and we have $\langle H,K\rangle =HK\cdots HK$ for a fixed number of factors. Maybe this property could be used in some way ?