Can some group $G$ have an infinite number of cosets? My textbook states the number of cosets of $H$ in a finite group $G$ is $|G|/|H|$, however Is it possible for a group $G$ to have a subgroup $H$ such that $gH$ has infinite cardinality? I can't think of any examples.
Edit, a more concise way of putting it is,
is there any $H\le G$ s.t. $|G/H|$ is infinite?

In the Euclidean plane with vector addition a line through the origin is a subgroup. Its cosets are the lines parallel to that line.
