If $f$ is a nonzero complex-valued function defined on a finite abelian group $A$ and $\hat f$ is its Fourier transform, then $|f|| \hat f| \ge |A|$, where $f$ and $\hat f$ 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 Chebotarev on complex roots of unity, and we thereby obtain a new proof of Chebotarev’s theorem.