
Journal of Lie Theory 29 (2019), No. 1, 221225 Copyright Heldermann Verlag 2019 On the Component Factor Group G/G_{0} of a ProLie Group G Rafael Dahmen Karlsruher Institut für Technologie, HermannvonHelmholtzPlatz 1, 76344 KarlsruheEggenstein, Germany rafael.dahmen@kit.edu Karl H. Hofmann Fachbereich Mathematik, Technische Universität, Schlossgartenstrasse 7, 64289 Darmstadt, Germany hofmann@mathematik.tudarmstadt.de A proLie group G is a topological group such that G is isomorphic to the projective limit of all quotient groups G/N (modulo closed normal subgroups N) such that G/N is a finite dimensional real Lie group. A topological group is almost connected if the totally disconnected factor group G_{t} = G/G_{0} of G modulo the identity component G_{0} is compact. In this case it is straightforward that each Lie group quotient G/N of G has finitely many components. However, in spite of a comprehensive literature on proLie groups, the following theorem, proved here, was not available until now: Theorem. A proLie group G is almost connected if each of its Lie group quotients G/N has finitely many connected components. The difficulty of the proof is the verification of the completeness of G_{t}. Keywords: ProLie groups, almost connected groups, projective limits. MSC: 22A05, 22E15, 22E65, 22E99 [ Fulltextpdf (78 KB)] for subscribers only. 