Let $H$ be a subgroup of a group $G$ and $x$ be an element of $G$.


If $G$ is finite, $xH\subset Hx$ implies $xH=Hx$ since $xH$ and $Hx$ have the same finite cardinality (that of $H$).


What happens when $G$ and $H$ are infinite? Can a left coset be strictly included in a right coset?



Does a left coset inclusion in a right coset imply they are equal?
In the free group on two generators $x, y$, if we take $H$ to be the subgroup generated by elements of the form $x^n y^m x^{-n}$ for $n \geq 0$, then $x H x^{-1}$ is strictly contained in $H$ because $y \notin x H x^{-1}$. 


