# Definition:Index of Subgroup

## Definition

Let $G$ be a group.

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

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

### Finite Index

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

### Infinite Index

If $G / H$ is an infinite set, then $\index G H$ is infinite, and $H$ is of infinite index in $G$.

## Also denoted as

Some sources use the notation $\card {G : H}$, and others use $\paren {G : H}$.

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

## Also defined as

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

Some, while developing the groundwork of the subject, refer to the left index and right index, according to whether the cardinality of the left coset space or right coset space is under consideration.

However, from Left and Right Coset Spaces are Equivalent, it follows that the left index and right index are in fact the same thing, and such a distinction is of minimal relevance.

## Also see

• Results about the index of subgroups can be found here.