Definition:Index of Subgroup

Definition
Let $G$ be a group.

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

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

Also denoted as
Some sources use the notation $\left\vert{G : H}\right\vert$, and others use $\left({G : H}\right)$.

Some merely use the notation for the cardinality of the coset space and write $\left\vert{G / H}\right\vert$.

Also defined as
Some sources define the index of a subgroup only for the case where $G$ is finite.

Also see

 * Left and Right Coset Spaces are Equivalent, demonstrating that this definition is meaningful.