If $(aH)(bH) = cH$, then $cH = abH$. Let $G$ be a group and $H$ a subgroup, if $(aH)(bH) = cH$ for some $c\in G$, for all $a,b\in G$, then I want to show that $(aH)(bH) = abH$.
I know that $aH = bH$ iff $a^{-1}b \in H$.

Since the identity $e\in G$ is in $H$, if for $a,b,c$ we happened to have
$$(aH)(bH)=cH\,,$$
then, by $e\in H$, we see that $ab\in cH$, i.e. $ab=ch$ for some $h\in H$, then $cH=abh^{-1}H=abH$.
