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

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.

• See Wikipedia: the fundamental group of $E_6$ is $\Bbb Z/3\Bbb Z$. Sep 19 at 19:19

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 $$E_6$$.

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.