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Is it true that given a connected locally compact group $G$, there must be abelian subgroups $H_{1},\dots, H_{n}$ and a compact subgroup $K$ of $G$ such that $G$ is homeomorphic to $H_{1}\times H_{2}\times\cdots\times H_{n}\times K$?

If so, can anyone supply a reference?

Any response is greatly appreciated!

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  • $\begingroup$ What if I only asked the map $(h_{1},\dots,h_{n},k)\mapsto h_{1}\cdots h_{n}k$ to be a homeomorphism. Are you saying that would work? $\endgroup$ – roo Nov 24 '16 at 9:16
  • $\begingroup$ It's not obvious to me that if you changed the direct product to semi-direct product, that there wouldn't be such subgroups of the new group which satisfy the statement. $\endgroup$ – roo Nov 24 '16 at 15:06
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Yes.

For every connected locally compact group $G$ there is a maximal compact subgroup $K$, and there exists $d\in\mathbf{N}$ (unique) such that $G$ is homeomorphic to $\mathbf{R}^d\times K$. See https://mathoverflow.net/a/140638/14094

So to conclude, it is enough to show that if $G$ is not compact, then it has a closed subgroup isomorphic to $\mathbf{R}$, and this is standard.

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