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.
1
Karlsruher Institut für Technologie, Hermann-von-Helmholtz-Platz 1, 76344 Karlsruhe-Eggenstein, Germany
2
Fachbereich Mathematik, Technische Universität, Schlossgartenstrasse 7, 64289 Darmstadt, Germany
Rafael Dahmen; Karl H. Hofmann. On the Component Factor Group G/G0 of a Pro-Lie Group G. Journal of Lie Theory, Tome 29 (2019) no. 1, pp. 221-225. http://geodesic.mathdoc.fr/item/JOLT_2019_29_1_a8/
@article{JOLT_2019_29_1_a8,
author = {Rafael Dahmen and Karl H. Hofmann},
title = {On the {Component} {Factor} {Group} {G/G\protect\textsubscript{0}} of a {Pro-Lie} {Group} {G}},
journal = {Journal of Lie Theory},
pages = {221--225},
year = {2019},
volume = {29},
number = {1},
url = {http://geodesic.mathdoc.fr/item/JOLT_2019_29_1_a8/}
}
TY - JOUR
AU - Rafael Dahmen
AU - Karl H. Hofmann
TI - On the Component Factor Group G/G0 of a Pro-Lie Group G
JO - Journal of Lie Theory
PY - 2019
SP - 221
EP - 225
VL - 29
IS - 1
UR - http://geodesic.mathdoc.fr/item/JOLT_2019_29_1_a8/
ID - JOLT_2019_29_1_a8
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%U http://geodesic.mathdoc.fr/item/JOLT_2019_29_1_a8/
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