Closed subgroup of finite index

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This article defines a property that can be evaluated for a subgroup of a semitopological group

Definition

A subgroup of a topological group (or more generally any of the variations of topological group that involve a group structure and a topological space structure, including left-topological group, right-topological group, semitopological group, quasitopological group, or paratopological group) is termed a closed subgroup of finite index or open subgroup of finite index if it satisfies the following equivalent conditions:

  1. It is a closed subgroup that is also a subgroup of finite index in the whole group
  2. It is an open subgroup that is also a subgroup of finite index in the whole group

Equivalence of definitions