# Torsion-free Lie groups

Out of curiosity, I'm looking for some examples of Lie groups that are torsion-free.

For some reason (and perhaps there is a good reason), most torsion-free groups I've heard of are discrete. One Lie example that comes to mind is $$\mathbb{R}^n$$. Also, if $$G$$ is a Lie group and its torsion subgroup, call it $$\Gamma$$, is both normal and discrete, then $$G/\Gamma$$ provides another source of examples.

Question 1: What are some more examples of torsion-free Lie groups?

Question 2: Are there any examples that have non-trivial compact Lie subgroups?

• The Heisenberg manifold is another example of torsion-free Lie group. Mar 2 at 9:20

The group of real upper triangular matrices with $$>0$$ diagonal entries is a good example of a group without elemnt of finite order.
If a Lie group contains a compact group , this subgroup is a compact Lie group, as every closed subgroup of a Lie group is a Lie group (Chevalley). But a compact Lie group is either finite (hence torsion) or contains a maximal torus, hence $$T^1$$, the group of complex number of modulus 1, and therefore contains elemnets of order $$n$$ for every $$n$$.