Category:Definitions/Index of Subgroups

This category contains definitions related to Index of Subgroups.
Related results can be found in Category:Index of Subgroups.

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 $G$.