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On closed abelian subgroups of real Lie groups. (English) Zbl 0879.22003
The author considers the structure of closed abelian and soluble subgroups of real Lie groups. A locally compact abelian group is called an elementary group if \(n\) \(m\) \(k\) is isomorphic to \(R\times T\times Z\times F\) where \(n,m,k\) are natural numbers and \(F\) is a finite subgroup. The main results are the following. Theorem 3.1. Let \(G\) be a connected real Lie group and \(A\) a closed abelian subgroup of \(G\). Then \(A\) is elementary. Theorem 3.2. Let \(G\) be a locally compact real Lie group and \(G\) the 0-identity component of \(G\). If the abelian subgroups of \(G/G\) are finitely 0-generated then the closed abelian subgroups of \(G\) are elementary. Theorem 3.5. Let \(G\) be a connected real Lie group and \(S\) be a closed soluble subgroup of \(G\). Then \(S\) is compactly generated. Theorem 3.6. Let \(G\) be a locally compact real Lie group and \(G\) the identity component of \(G\). If abelian subgroups of \(G/G\) are finitely 0-generated, then the closed soluble subgroups are compactly generated.

MSC:
22E15 General properties and structure of real Lie groups
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