Definition:Index of Subgroup

Definition
Let $G$ be a group.

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

The index $\left[{G : H}\right]$ of $H$ (in $G$) is the number of left (or right) cosets of $G$ modulo $H$, or, the number of elements in the left (or right) coset space $G / H$, provided this number is finite.

Otherwise, the index is said to be infinite.

Also see

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

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