#### Vol. 15, No. 1, 1965

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Character sums and difference sets

### Richard Joseph Turyn

Vol. 15 (1965), No. 1, 319–346
##### 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 $\chi$ any character of $G$, $\chi \left(D\right)={\sum }_{D}\chi \left(g\right)$ is an algebraic integer of absolute value $\sqrt{n}$ in the field of $m$-th roots of 1, where $m$ is the order of $\chi$. Known facts about such integers and the relations which the $\chi \left(D\right)$ must satisfy (as $\chi$ varies) may yield information about $D$ by the Fourier inversion formula. In particular, if $\chi \left(D\right)$ is necessarily divisible by a relatively large integer, the number of elements $g$ of $D$ for which $\chi \left(g\right)$ 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 ${2}^{a}$, 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.

Primary: 05.22
Secondary: 10.41
##### Milestones
Received: 17 March 1964
Published: 1 March 1965
##### Authors
 Richard Joseph Turyn