Inequalities for Finite Group Permutation Modules

If f is a nonzero complex-valued function defined on a finite abelian group A and f̂ is its Fourier transform, then | supp(f)|| supp(f̂)| ≥ |A|, where supp(f) and supp(f̂) are the supports of f and f̂ . 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.