  If f is a nonzero complex-valued function defined on a finite abelian group A
and \hat f is its Fourier transform, then |Supp (f)||Supp {\hat f)| \ge |A|,
where Supp (f) and Supp (\hat f) are the supports of f and \hat f. In this
paper we generalize this known result in several directions. In particular, we
prove an analogous inequality where the abelian group A is replaced by a
transitive right G-set, where G is an arbitrary finite group. We obtain
stronger inequalities when the G-set is primitive and we determine the
primitive groups for which equality holds. We also explore connections between
inequalities of this type and a result of Chebotar\"ev on complex roots of
unity, and we thereby obtain a new proof of Chebotar\"ev's theorem.
