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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:
 2.2e+16 General properties and structure of real Lie groups
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