A Wilbrink-like equation for neo-difference sets (Q1653309)

Let \(G\) be a group of order \((n-1)^{2}\) with three pairwise trivally intersecting subgroups \(X\), \(Y\), \(Z\). A subset \(D\subseteq G\) is called a neo-difference set of order \(n\) with forbidden subgroups \(X\), \(Y\), \(Z,\) if every element \(g\in G\setminus (X\cup Y\cup Z)\) has a unique representation \(g=d_{1}d_{2}^{-1}\) with \(d_{i}\in D\), but elements inside \(X\cup Y\cup Z\) do not. Notice that \(|D|=n-2\). Each subset of \(G\) can be represented in the integral group ring \(ZG\) of \(G\) in an obvious way. Thus, eg., \(D=\sum_{g\in D} g\); and \(D^{(-1)}:=\sum_{g\in D} g^{-1}\). Being a neo-difference set is equivalent to \(DD^{(-1)}=n+G-( X+ Y+ Z)\). The authors prove: If \(G\) is abelian, and \(D\) has \(pm\) elements, with \(p\) an odd prime and \(p\nmid m\), then \[ D^{p-1}+(D^{(-1)})^{p-1}=1-2G-( X+ Y+ Z). \] Neo-difference sets arise in the study of finite projective planes of Lenz-Barlotti classes I.3 and I.4, which are conjectured to be non-existent. After having consulted the authors, the reviewer would like to point out two typos in the paper: In the presumptions, third point: the set \(U_{3}\) must be \(U_{3}=\{(x,x): x\in X\}\). In Corollary 2.3(iii), the hypothesis should be \(A\in ZH\), rather than \(A\in ZG\).