  We formulate a version of the Pompeiu problem in the discrete group setting.
Necessary and sufficient conditions are given for a finite collection of finite
subsets of a discrete abelian group, whose torsion free rank is less than the
cardinal of the continuum, to have the Pompeiu property. We also prove a
similar result for nonabelian free groups. A sufficient condition is given that
guarantees the harmonicity of a function on a nonabelian free group if it
satisfies the mean-value property over two spheres.
