Nonexistence of abelian difference sets: Lander's conjecture for prime power orders

Abstract

In 1963 Ryser conjectured that there are no circulant Hadamard matrices of order > 4 and no cyclic difference sets whose order is not coprime to the group order. These conjectures are special cases of Lander's conjecture which asserts that there is no abelian group with a cyclic Sylow p-subgroup containing a difference set of order divisible by p. We verify Lander's conjecture for all difference sets whose order is a power of a prime greater than 3.