Definition:Index of Subgroup/Infinite

Definition
Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Let $\left[{G : H}\right]$ denote the index of $H$ in $G$, that is, the cardinality of the left (or right) coset space $G / H$.

If $G / H$ is an infinite set, then $\left[{G : H}\right]$ is infinite, and $H$ is of infinite index in $G$.

Also see

 * Definition:Finite Index