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The purpose of this article is to study the fundamental group of the polyhedral product denoted $Z_K(X,A)$, which denotes the moment-angle complex of Buchstaber-Panov in the case $(X,A) = (D^2, S^1)$, with extension to arbitrary pairs in [2] as given in Definition 2.2 here.\n  For the case of a discrete group $G$, we give necessary and sufficient conditions on the abstract simplicial complex $K$ such that the polyhedral product denoted by $Z_K(\\underline{BG})$ is an Eilenberg-Mac Lane space. 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