Let $G$ be any group, and $H \leq G$ a subgroup.
Suppose that for each $x \in G$, there exists a $y \in G$ such that $xH \subseteq Hy$. In other words, every left coset of $H$ is contained inside some right coset of $H$.
Question: what can we say about $H$? In particular, does this imply that $H$ is normal?
I know that if $G$ is finite, then $H$ must be normal, because then $xH \subseteq Hy$ implies $xH = Hy$, since $|xH| = |Hy|$.