Definition:Index of Subgroup/Finite

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 a finite set, then $\index G H$ is finite, and $H$ is of finite index in $G$.

Also see

 * Definition:Infinite Index