Connected subgroups of Lie groups (Q1087654)

A subgroup H of a Lie group is said to be an analytic subgroup if H is a connected Lie group and the inclusion map is an immersion. It has long been known that H is an analytic subgroup if and only if it is pathwise connected. In this note we show that pathwise connectivity cannot be weakened to connectivity. The precise result is: Theorem. Let G be a connected Lie group of dimension greater than one. Then G has a connected subgroup which is dense in G and which is not an analytic subgroup.