# Definition:Index of Subgroup

*This page is about Index of Subgroup. For other uses, see index.*

## 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

Other notations seen for the **index** $\index G H$ of a subgroup $H$ in a group $G$ include:

- $\card {G : H}$
- $\paren {G : H}$
- $\operatorname {index} H$ (which is not recommended, as it does not indicate the group of which $H$ is a subgroup)

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

## Also defined as

Some sources define the **index $\index G H$ 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

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

- Results about
**the index of subgroups**can be found**here**.

## Linguistic Note

The plural of **index** is **indices**.

Compare **vertex** and **apex**, which have a similar plural form.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 6.3$. Index. Transversals - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.9$ - 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): Introduction: Notation - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: Problem $\text{GG}$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 39$ - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 43$. Lagrange's theorem - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Definition $5.10$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**index**(*plural***indices**)**${}$**:**4.**(of a subgroup)