Journal of Lie Theory 18 (2008), No. 2, 383--390
Copyright Heldermann Verlag 2008
On the Pro-Lie Group Theorem and the Closed Subgroup Theorem
Karl H. Hofmann
Fachbereich Mathematik, Technische Universitšt, Schlossgartenstr. 7, 64289 Darmstadt, Germany
Sidney A. Morris
School of Information Technology and Mathematical Sciences, University of Ballarat, P. O. Box 663, Ballarat, Vic. 3353, Australia
Let $H$ and $M$ be closed normal subgroups of a pro-Lie group $G$ and assume that $H$ is connected and that $G/M$ is a Lie group. Then there is a closed normal subgroup $N$ of $G$ such that $N\subseteq M$, that $G/N$ is a Lie group, and that $HN$ is closed in $G$. As a consequence, $H/(H\cap N)\to HN/N$ is an isomorphism of Lie groups.
Keywords: Pro-Lie groups, closed subgroup theorem.
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