Definition:Index of Subgroup

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 infinite).

Notation
Some sources use the notation $$\left|{G : H}\right|$$.