I'm searching for a good reference that prove the descending chain propriety for compact Lie groups (i.e. every sequence $K_1\supset K_2\supset...$ of closed subgroups $K_i$ of $G$ is eventually constant).
Thank you!
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Am I missing something, or does it follow from the fact that a closed subgroup is either a union of connected components or of lower dimension, and by compactness the number of connected components is finite, while by Lieness, the dimension is finite?
answered Jul 3, 2018 at 18:02
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$\begingroup$ Yes! What I have not found is a proof of the fact that a closed subgroup is either a union of connected components or of lower dimension. Thanks! $\endgroup$ Commented Jul 3, 2018 at 18:12
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4$\begingroup$ The follows from the fact that a closed subgroup is a Lie subgroup, and so its identity component is determined by its Lie algebra. $\endgroup$ Commented Jul 3, 2018 at 18:17
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$\begingroup$ Thank you @BenMcKay! Is my argumentation correct? If I have H closed subgroup of G then H is a submanifold. Now I have two possibilities: dim(H)=dim(G) or dim(H)<dim(G). If dim(H)=dim(G) then H is also an open submanifold of G (is it true that it simply follows from the definition of submanifold?), thus H has less connected components than G. $\endgroup$ Commented Jul 4, 2018 at 9:56
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