Sharply flag-transitive projective planes and power residue difference sets

The study of sharply flag-transitive projective planes leads to a particular case of the class of power residue difference sets. In this paper we are concerned with the existence of such difference sets, which means that we ask for conditions whether the set of d-powers or the set of d-powers together with the zero element in a finite field yields a difference set. By using Jacobi sums we prove that d is in fact a power of 2. As a corollary we obtain the result that the order of a finite sharply flag-transitive projective plane is a power of 2. (C) 2004 Elsevier Inc. All rights reserved.