When are finite projective planes magic?

TL;DR

This paper explores the conditions under which finite projective planes can be labeled with Abelian groups to exhibit a 'magic' property, providing classifications especially for prime order planes.

Contribution

It introduces a generalization of magic squares to finite projective planes using Abelian groups and classifies all such groups for prime order planes.

Findings

Finite projective planes can be labeled with Abelian groups to be 'magical'.

Complete classification of groups for prime order planes.

Demonstrates the existence of small groups for labeling.

Abstract

This article studies a generalization of magic squares to finite projective planes. In traditional magic squares the entries come from the natural numbers. This does not work for finite projective planes, so we instead use Abelian groups. For each finite projective plane we demonstrate a small group over which the plane can labeled magically. In the prime order case we classify all groups over which the projective plane can be made magic.