New necessary conditions on the existence of Abelian difference sets (Q1193535)

It is still an unsolved conjecture that the order \(n\) of a cyclic (or abelian) finite projective plane (i.e. a plane with a cyclic or abelian automorphism group \(G\) acting sharply transitively on the points) has to be a prime power. \textit{H. A. Wilbrink} [J. Comb. Theory, Ser. A 38, 94-95 (1985; Zbl 0554.05012)] proved an interesting algebraic equation for the corresponding difference set \(D\) in \(\text{GF}(p)G\) provided that the order \(n\) is divisible by \(p\) but not by \(p^ 2\). One can use Wilbrink's equation to get evidence for the above mentioned prime power conjecture. This paper is an attempt to weaken the assumption on \(p\). We derive equations in \(\mathbb{Z}_{p^ i}G\) (using the group ring over the \(p\)-adic number field) if \(p^ i\) is the exact divisor of \(n\). Unfortunately, the equations are at present too complicated to have interesting applications.