An uncertainty inequality for groups of order pq

Abstract

We are concerned with some relations between the support of a function defined on a finite group and the support of its Fourier transform. Let D p,q be the non-commutative group of order pq , where p, q are primes, and p | q −1. We prove an uncertainty-type inequality for D N p,q = {( x 1 , …, x N ): x i ∈ D p,q }, which has the following application: If H is a subgroup of D N p,q and x 1 … x N = 1 for all ( x 1 , …, x N ) ∈ H , then ( D N p,q : H ) ⩾ q ½ p ( N −1)/2 .