The aim of this paper is to study the structure of the fundamental group of a closed oriented Riemannian manifold with positive scalar curvature. To be more precise, let M be a closed oriented Riemannian manifold of dimension n (4 $leq$ n $leq$ 7) with positive scalar curvature and non-trivial first Betti number, and let be $alpha$ non-trivial codimension one homology class in $H_{n-1}$(M;$mathbb{R}$). Then it is known as in [8] that there exists a closed embedded hypersurface $N_{alpha}$ of M representing $alpha$ of minimum volume, compared with all other closed hypersurfaces in the homology class. Our main result is to show that the fundamental group ${pi}_1(N_{alpha})$ is always virtually free. In particular, this gives rise to a new obstruction to the existence of a metric of positive scalar curvature.