Definition:Index of Subgroup/Infinite
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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
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