|
This paper concerns difference sets in finite groups. The approach is as follows: if
is a difference
set in a group
,
and
any
character of
,
is an algebraic integer
of absolute value
in the field of
-th
roots of 1, where
is the order of
.
Known facts about such integers and the relations which the
must satisfy (as
varies) may yield
information about
by the Fourier inversion formula. In particular, if
is
necessarily divisible by a relatively large integer, the number of elements
of
for
which
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
and
are both even and
, the power of 2 in
, is at least half
of that in
, then
cannot have a
character of order
,
and thus
cannot be cyclic.
A difference set with
gives rise to an Hadamard matrix; it has been conjectured that no such cyclic sets exist with
. This is
proved for
even by the above theorem, and is proved for various odd
by the theorems which depend on magnitude arguments. In the last
section, two classes of abelian, but not cyclic, difference sets with
are
exhibited.
|
