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.