# Definition:Index of Subgroup/Infinite

## Definition

Let $G$ be a group.

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

Let $\index G H$ 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 $\index G H$ is infinite, and $H$ is of infinite index in $G$.