Is it possible that a left coset of $H$ contains more than one right coset of $H$? Let $H$ be a subgroup of group $G$. 

Is it possible that a left coset of $H$ contains more than one right coset of $H$?

It is clear to me that the answer is 'no' if we deal with finite groups.

Yes, it's possible.
For example, let $G=\langle x,y: y^{-1}xy=x^2\rangle$ and $H=\langle x^2\rangle$.
Then $\langle x\rangle = H\cup Hx$, so $H=y^{-1}Hy\cup y^{-1}Hxy$, and so $yH=Hy\cup Hxy$.
You'll get a similar example from any group $G$ with a subgroup $H$ that is properly contained in some conjugate of $H$.
