Is it true that the class of isomorphism classes of fundamental groups of Lie groups coincides with the class of isomorphism classes of finitely generated abelian groups?

## 1 Answer

Yes, these two classes coincide. One direction splits into two: for a Lie group $G$,

*the fundamental group is abelian*: this is already answered here this MO question, and*the fundamental group is finitely generated*. This is equivalent (passing to the universal covering of the unit component) to the property that every discrete central subgroup of a connected Lie group is finitely generated, and this is answered here.

Conversely any finitely generated abelian group is fundamental group of a Lie group. By the classification of finitely generated abelian groups, it is enough to consider cyclic groups (and then use products). Indeed $\mathbf{Z}$ is isomorphic to the fundamental group of the circle group $\mathbf{R}/\mathbf{Z}$, and $\mathbf{Z}/n\mathbf{Z}$ is isomorphic to the fundamental group of $\mathrm{PSL}_n(\mathbf{C})$.

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