Yes, if $G$ is an abelian group, its classifying space $BG$ is an abelian topological group whose $\pi _1$ is $G$. You can find details in Joan Baez's wonderful (as usual) post http://math.ucr.edu/home/baez/calgary/BG.html
or in the answer by Chris Schommer-Priess to the following question on this site: Classifying Space of a Group Extension