Connected implies no proper open subgroup

From Groupprops
Revision as of 22:31, 14 January 2012 by Vipul (talk | contribs) (Related facts)
Jump to: navigation, search

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

Similar facts

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:

Facts used

  1. Open subgroup implies closed

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.