$p$-torsion in fundamental group of Lie group.

Question

I'm study theory of Lie groups and the question is: can Lie group G have $p$-torsion for prime $p \neq 2$ in its fundamental group $π_1(G)$?

I know standart examples of Lie groups like $\mathrm{O}_n(ℝ), \mathrm{SL}_n(ℝ), \mathrm{GL}_n(ℂ)$ ... , but their fundamental groups are equal to one of $1, ℤ, ℤ_2$ (or direct sum of them).

Of course $H_1(G, ℤ) ≃ π_1(G)$, but it didn’t help me.

Thanks in advance.

Answer 1

First, the fundamental group of a connected Lie group is abelian and finitely generated.

Secondly, any finitely generated abelian group is the fundamental group of a connected Lie group.

To see the second claim, note that by the classification of finitely generated abelian groups, it is enough to consider cyclic groups (and then use products). Indeed $\Bbb Z$ is isomorphic to the fundamental group of the circle group $\mathbb{R}/\mathbb{Z}$, and $\mathbb{Z}/n\mathbb{Z}$ is isomorphic to the fundamental group of $\mathrm{PSL}_n(\mathbb{C})$.

For simple Lie groups, we can also have $\Bbb Z/3\Bbb Z$ as fundamental group, e.g., taking the complex Lie group $E_6$.

Reference: https://mathoverflow.net/questions/294779/fundamental-group-of-a-lie-group

Answer 2

It's a fundamental result that if $G$ is a connected Lie group then the universal covering map $\widetilde{G} \to G$ has kernel a central subgroup of $\widetilde{G}$ isomorphic to $\pi_1(G)$, giving a canonical short exact sequence

$$1 \to \pi_1(G) \to \widetilde{G} \to G \to 1$$

exhibiting $\widetilde{G}$ as a central extension of $G$ by $\pi_1(G)$. The upshot is that to exhibit a (finitely generated) abelian group $\pi$ as the fundamental group of a Lie group it suffices to find a simply connected Lie group in which $\pi$ sits as a central subgroup.

Now we just observe that, for example, the center of $SU(n)$ is $\mathbb{Z}/n\mathbb{Z}$, so central quotients of $SU(n)$ (the maximal one of which is $PSU(n) \cong SU(n)/Z(SU(n))$ which satisfies $\pi_1(PSU(n)) \cong \mathbb{Z}/n\mathbb{Z}$) have $p$-torsion in their fundamental group for suitable $n$. $SU(n)$ can also be replaced by $SL_n(\mathbb{C})$ if you prefer algebraic groups to compact groups.