Character sums and difference sets

Abstract

This paper concerns difference sets in finite groups. The approach is as follows: if D is a difference set in a group G, and χ any character of G, χ(D) = ∑pχ(g) is an algebraic integer of absolute value in the field of mth roots of 1, where m is the order of χ. Known facts about such integers and the relations which the χ(D) must satisfy (as χ varies) may yield information about D by the Fourier inversion formula. In particular, if χ(D) is necessarily divisible by a relatively large integer, the number of elements g of D for which χ(g) takes on any given value must be large; this yields some nonexistence theorems. Another theorem, which does not depend on a magnitude argument, states that if n and v are both even and a, the power of 2 in v, is at least half of that in n, then G cannot have a character of order 2a, and thus G cannot be cyclic. A difference set with v = 4n gives rise to an Hadamard matrix; it has been conjectured that no such cyclic sets exist with v > 4. This is proved for n even by the above theorem, and is proved for various odd n by the theorems which depend on magnitude arguments. In the last section, two classes of abelian, but not cyclic, difference sets with v = 4n are exhibited. © 1965 by Pacific Journal of Mathematics.