Connected implies no proper open subgroup: Difference between revisions
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==Statement== | ==Statement== | ||
===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]]. | |||
===Statement for topological groups=== | |||
A [[fact about::connected topological group;1| ]][[connected topological group]] has no [[proper subgroup|proper]] [[fact about::open subgroup;1| ]][[open subgroup]]. | A [[fact about::connected topological group;1| ]][[connected topological group]] has no [[proper subgroup|proper]] [[fact about::open subgroup;1| ]][[open subgroup]]. | ||
Revision as of 22:21, 14 January 2012
Statement
Statement for semitopological groups
A connected semitopological group has no proper open subgroup.
Statement for topological groups
A connected topological group has no proper open subgroup.
Related facts
Converse
The converse is not true for all groups, i.e., it is possible to have a group that is not connected but has no proper open subgroup. An example is the additive group of rational numbers with the Euclidean topology -- this is a totally disconnected group but has no proper open subgroup.
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
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.