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Let $G$ be a real (finite-dimensional) connected Lie group. Suppose that $H \subset G$ is a connected Lie subgroup, and $\dim(H) = \dim(G)$, does it imply that $H = G$?

If not, what would be a simple counterexample? Any mild additional assumption would make this hold? Thanks!

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Note that since $dim(H) = dim(G)$ the inclusion $H \hookrightarrow G$ is an open map, therefore $H$ is open in $G$. We know that the identity is in $H$, but for any open set containing the identity, the group genrated by it is the whole group (G is connected), thus $H=G$.

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    $\begingroup$ Nice. Thanks a lot. $\endgroup$
    – B Chung
    Mar 28 '21 at 0:24

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