Example: Lie group compact, abelian and disconnected.

Question

I'm looking for a example of a Lie group compact, abelian and disconnected, such that exist some elements $x$, where $x^n\neq e, \forall n\in\mathbb{N}$.

Answer 1

Try irrational rotations in $U(1)\times (\mathbb{Z}/2\mathbb{Z})$. It forms such an example:

• It is the product of two abelian groups, hence abelian
• Topologically, it is a disjoint union of two copies of $S^1$, hence compact and disconnected
• Irrational rotations give infinite cyclic subgroups