Definition:Index of Subgroup/Finite

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

Also see

• Definition:Infinite Index

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.3$. Index. Transversals
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 39$
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 5$: Groups $\text{I}$: Subgroups
• 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