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$G$ is an algebraic group, and $H$ is a subgroup which is solvable. $\overline{H}$ is its closure in $G$.

Then $\overline{H}$ is also a subgroup of $G$. Is it also solvable?

For any algebraic group $G$, denote $[G,G]$ the derived subgroup of $G$. Then is it true that $\overline{[H,H]} = [\overline{H}, \overline{H}]$? If this is true, I think the solvability of $\overline{H}$ might be proved by dimension comparision.

Many thanks~ Special thanks to @awllower for the enlightening comments.

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    Any closure of a subgroup is also a subgroup, for the first part.2011-11-23
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    @awllower: Thank you very much for your comment~ It is true that the closure of a subgroup is still a subgroup. I think for any subgroup $H$ of $G$, and the closure $\overline{H}$ of $H$, if it is true that the closure of the derived group of $H$ is just the derived group of $\overline{H}$, then I can prove from the solvability of $H$ that $\overline{H}$ is solvable by dimension comparision. But I don't know the correctness of $\overline{(H,H)}= (\overline{H}, \overline{H})$...2011-11-24
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    Indeed I was wondering if there is some way we can relate the derived subgroups and the closures of them; so far no much progress is in hand. Sorry I cannot provide an answer. Does it make much difference to work with algebraic groups, from working with just topological groups? Maybe you can change the tag? Thanks for listening.2011-11-25
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    @awllower: Thank you very much for the enlightment. Algebraic group is a special type of topological groups, so similar results on topological groups in general may shed light on this problem. I will edit the tag :)2011-11-25
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    Thanks for your generous words.2011-11-27
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    @awllower: I am afraid algebraic groups are not topological groups because the topology given on product groups are different, the former Zariski topology and the latter product topology. I posted another question and hopefully it will get some attention. http://math.stackexchange.com/questions/86128/for-a-topological-group-g-and-a-subgroup-h-is-it-true-that-overlineh Thanks for the enlightening discussion.2011-11-27

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