Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and $H(\mathbb{R})$ are compact (and connected). If $G(\mathbb{R})\cap H(\mathbb{R})$ is a subgroup defined over $\mathbb{Q}$, then what we can say about $G$ and $H$? Is it true that either $H$ is defined over $\mathbb{Q}$ or $G$ is contained in $H$?
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$\begingroup$ Crossposted on MSE. $\endgroup$– Michael AlbaneseCommented Apr 1, 2015 at 22:55
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$\begingroup$ Does "defined over $Q$" means "definable over Q"? when you say algebraic subgroups, do you mean that $G$ is an $Q$-subgroup? an $R$-subgroup? do you mean that $H$ is an $R$-subgroup? $\endgroup$– YCorCommented Apr 1, 2015 at 23:15
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$\begingroup$ Can't say anything; what if $G_{\mathbf{R}} \cap H = 1$? The hypotheses need to be made more specific. What is the motivation? $\endgroup$– user74230Commented Apr 2, 2015 at 1:00
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