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I am stuck on the following question

Let $T^n$ be a $n$-dimensional torus. What are the connected Lie subgroups of $T^n$? Which of these are closed subgroups?

Does anyone have any ideas on how to solve to this?

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What have you tried? Notice that the preimage under the map $R^n\to T^n$ of any connected subgroup will contain a connected subgroup of $R^n$, and this is a bijection... – Mariano Suárez-Alvarez Apr 15 '12 at 23:57
This is what I have tried. As $T^n$ is abelian it has a zero bracket. So all subspaces of the lie algebra $\mathfrak{g}$ of $T^n$ are lie sub-algebras. Can you then just say then that the set of connected lie subgroups is just $\{exp(A) \ : \ A \ \text{is a subspace of $\mathfrak{g}$} \}$ where exp is the exponential map? I really not sure not that this is true though. – Alex Kite Apr 16 '12 at 15:22

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