The isomorphism problem for abelian projective planes

Abstract

We settle the isomorphism problem for finite cyclic projective planes by proving the following analogue of the Bays–Lambossy theorem: two cyclic projective planes of order n are isomorphic if and only if they are multiplier equivalent, that is, if and only if the associated difference sets in $${\mathbb{Z}_{n^2+n+1}}$$ are equivalent. In fact, we establish a more general result for abelian groups, where multipliers are replaced by group automorphisms.