Connected implies no proper open subgroup: Difference between revisions
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===Statement for semitopological groups=== | ===Statement for semitopological groups=== | ||
A [[fact about::connected semitopological group;1| ]][[connected semitopological group]] has no [[proper subgroup|proper]] [[fact about::open subgroup;1| ]][[open subgroup]]. | A connected [[left-topological group]] has no [[proper open subgroup|proper]] [[open subgroup]]. Similarly, a connected right-topological group has no proper open subgroup. | ||
Since semitopological groups are both left-topological and right-topologica, this tells us that a [[fact about::connected semitopological group;1| ]][[connected semitopological group]] has no [[proper subgroup|proper]] [[fact about::open subgroup;1| ]][[open subgroup]]. | |||
===Statement for topological groups=== | ===Statement for topological groups=== | ||
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==Related facts== | ==Related facts== | ||
===Similar facts=== | |||
* [[Open subgroup implies closed]] | * [[Open subgroup implies closed]] | ||
* [[Closed subgroup of finite index implies open]] | |||
* [[Compact implies every open subgroup has finite index]] | |||
===Converse=== | ===Converse=== | ||
The converse is not true for all groups | The converse is not true for all groups. See [[no proper open subgroup not implies connected]]. | ||
However, the converse is true in some contexts: | However, the converse is true in some contexts: | ||
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==Facts used== | ==Facts used== | ||
# [[uses::Open subgroup implies closed]] | # [[uses::Open subgroup implies closed]] (this is true in both left-topological groups and right-topological groups) | ||
==Proof== | ==Proof== | ||
By Fact (1), a proper open subgroup is a nonempty subset that is both open and closed (note that it is nonempty because it is a subgroup). The existence of such a subset contradicts connectedness. | By Fact (1), a proper open subgroup is a nonempty subset that is both open and closed (note that it is nonempty because it is a subgroup). The existence of such a subset contradicts connectedness. | ||
Latest revision as of 23:26, 23 June 2012
Statement
Statement for semitopological groups
A connected left-topological group has no proper open subgroup. Similarly, a connected right-topological group has no proper open subgroup.
Since semitopological groups are both left-topological and right-topologica, this tells us that a connected semitopological group has no proper open subgroup.
Statement for topological groups
A connected topological group has no proper open subgroup.
Related facts
Similar facts
- Open subgroup implies closed
- Closed subgroup of finite index implies open
- Compact implies every open subgroup has finite index
Converse
The converse is not true for all groups. See no proper open subgroup not implies connected.
However, the converse is true in some contexts:
- It is true for algebraic groups, i.e.,it is true if the topology is a Zariski topology. See equivalence of definitions of connected algebraic group.
- It is true for all locally connected topological groups. In particular, it is true for Lie groups. See equivalence of definitions of connected Lie group.
Facts used
- Open subgroup implies closed (this is true in both left-topological groups and right-topological groups)
Proof
By Fact (1), a proper open subgroup is a nonempty subset that is both open and closed (note that it is nonempty because it is a subgroup). The existence of such a subset contradicts connectedness.