Inequalities for finite group permutation modules

If $f$ is a nonzero complex-valued function defined on a finite abelian group $A$ and $\hat f$ is its Fourier transform, then $\vert\operatorname{supp}(f)\vert\vert\operatorname{supp}({\hat f})\vert \ge \vert A\vert$, where $\operatorname{supp}(f)$ and $\operatorname{supp}({\hat f})$ 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 Chebotarëv on complex roots of unity, and we thereby obtain a new proof of Chebotarëv's theorem.