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?

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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