Closed subgroup of finite index: Difference between revisions
(Created page with "{{topological subgroup property}} ==Definition== A subgroup of a topological group is termed a '''closed subgroup of finite index''' or '''open subgroup of finite in...") |
No edit summary |
||
| Line 3: | Line 3: | ||
==Definition== | ==Definition== | ||
A [[subgroup]] of a [[topological group]] is termed a '''closed subgroup of finite index''' or '''open subgroup of finite index''' if it satisfies the following equivalent conditions: | 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: | ||
# It is a [[defining ingredient::closed subgroup]] that is also a [[defining ingredient::subgroup of finite index]] in the whole group | # It is a [[defining ingredient::closed subgroup]] that is also a [[defining ingredient::subgroup of finite index]] in the whole group | ||
Latest revision as of 23:22, 23 June 2012
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:
- It is a closed subgroup that is also a subgroup of finite index in the whole group
- It is an open subgroup that is also a subgroup of finite index in the whole group
Equivalence of definitions
- (1) implies (2): closed subgroup of finite index implies open
- (2) implies (1): follows from open subgroup implies closed