Definition:Index of Subgroup

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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

  • 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

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