# A subgroup such that every left coset is contained in a right coset.

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 $|xH| = |Hy|$.

-

If $xH \subseteq Hy$, then in particular $x \in Hy$. But then $Hx \cap Hy \neq \varnothing$, so $Hx=Hy$, so $xH \subseteq Hx$, so $xHx^{-1}\subseteq H$. Since this is true for every $x$, $H$ is normal.
If $xH \subseteq Hy$ for some $x, y \in G$ (not all), then can we still have $xH = Hy = Hx$? – Lao-tzu Dec 3 '15 at 2:44