  Let $K$ be a finite simplicial complex, and $(X,A)$ be a pair of spaces. 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.
  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. The
fundamental group of $Z_K(\underline{BG})$ is shown to depend only on the
1-skeleton of $K$. Further special examples of polyhedral products are also
investigated.
  Finally, we use polyhedral products to study an extension problem related to
transitively commutative groups, which are given in Definition 5.2.
