Let $G$ be a compact Lie group. A torus in $G$ is a subgroup $T \leq G$ isomorphic to $(S^1)^n$ for some $n \geq 0$, where we set $(S^1)^0 = \{*\}$ which is the trivial torus.

A standard theorem asserts the existence of a maximal torus $T$ in $G$, can it be trivial when $G$ has positive dimension? Why not?

  • 2
    $\begingroup$ The closure of an abelian subgroup is again an abelian subgroup. Does $G$ have nontrivial connected abelian subgroups? $\endgroup$ – Daniel Fischer Sep 27 '16 at 10:41
  • $\begingroup$ Sorry, I don't see how that helps. A trivial torus is closed because the Lie group is a Hausdorff space, isn't it? So starting with a trivial torus, I don't see why it has to be contained in a larger one. $\endgroup$ – hallborrey Sep 27 '16 at 11:00
  • $\begingroup$ Don't start with a torus, start with a nontrivial connected abelian subgroup. And look at this or this. Plus this. $\endgroup$ – Daniel Fischer Sep 27 '16 at 11:12

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