
Journal of Lie Theory 27 (2017), No. 2, 569578 Copyright Heldermann Verlag 2017 Compact Elements in Connected Lie Groups Mikhail Kabenyuk Institute of Fundamental Sciences, Kemerovo State University, 650043 Kemerovo, Russia kabenyuk@kemsu.ru We prove that the set of compact elements in the group extension of the 3dimensional Heisenberg group by SO(2) (the socalled oscillator group) is not dense. We also give a new proof of the following criterion: The set of compact elements of a connected Lie group G is dense in G if and only if every Cartan subgroup of G is compact. Keywords: Lie group, compact element, Heisenberg group, oscillator group, Cartan subgroup. MSC: 22C05, 22E15, 22E25 [ Fulltextpdf (218 KB)] for subscribers only. 