Definition:Index of Subgroup/Infinite

Definition

Let $G$ be a group.

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

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

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

Also see

• Definition:Finite Index

Sources

• 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