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?

An interesting fact about the Prüfer $p$-group $\mathbb{Z}(p^\infty)$: if $H$ is a proper subgroup of $\mathbb{Z}(p^\infty)$, then
$$
\mathbb{Z}(p^\infty)/H\cong \mathbb{Z}(p^\infty)
$$
However, all proper subgroups of $\mathbb{Z}(p^\infty)$ are finite.
For other examples, consider $V$ a two-dimensional vector space over an infinite field and $U$ a one-dimensional subspace; then $U$ is infinite and $V/U$ is infinite as well.
