On Sylow subgroups of abelian affine difference sets (Q5929719)

Let \(G\) be an abelian group of order \(n^2-1\) containing an affine difference set \(R\) of order \(n\). This is a subset of \(G\) such that the list of differences \(d-d'\) with \(d,d'\in R\), \(d\neq d'\), covers every element outside a certain subgroup of order \(n-1\) precisely once. These objects correspond to projective planes with a quasiregular collineation group. They exist in cyclic groups whenever \(n\) is a prime power. It is conjectured that these are the only cases. In particular, the Sylow subgroups of \(G\) have to be cyclic. The paper under review gives bounds on the rank of the Sylow \(p\)-subgroups that hold under some technical assumptions on \(p\) and \(n\).