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.
