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Journal of Lie Theory 29 (2019), No. 1, 221--225
Copyright Heldermann Verlag 2019

On the Component Factor Group G/G0 of a Pro-Lie Group G

Rafael Dahmen
Karlsruher Institut für Technologie, Hermann-von-Helmholtz-Platz 1, 76344 Karlsruhe-Eggenstein, Germany

Karl H. Hofmann
Fachbereich Mathematik, Technische Universität, Schlossgartenstrasse 7, 64289 Darmstadt, Germany

A pro-Lie 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 Gt = G/G0 of G modulo the identity component G0 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 pro-Lie groups, the following theorem, proved here, was not available until now:
Theorem. A pro-Lie 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 Gt.

Keywords: Pro-Lie groups, almost connected groups, projective limits.

MSC: 22A05, 22E15, 22E65, 22E99

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