# On compact, simply connected Lie group and its subgroup

Let $G$ be a compact, simply connected Lie group, and $H$ is its Lie subgroup that is also compact and simply connected, and has the same dimension with $G$, then should $H=G$? Note that these imply that the Lie group $G$ and $H$ are locally isomorphic (consider a small neighborhood around the identity).

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What happens if $H=\left\{ 1_G \right\}$? –  busman Nov 18 '13 at 12:00
Sorry, I've modified my question just now. –  Lao-tzu Nov 18 '13 at 12:01