Set of involutions arising from an egglike inversive plane

Abstract

To every egglike inversive plane \( \Omega \) there is associated a family \( \mathcal{F}(\Omega) \) of involutions of the point set of \( \Omega \) such that circles of \( \Omega \) are the fixed point sets of the involutions in \( \mathcal{F}(\Omega) \). Korchmaros and Olanda characterized a family \( \Omega \) of involutions on a set of size n2 + 1to be \( \mathcal{F}(\Omega) \) for an egglike inversive plane of order n by four conditions. In this paper, we give an alternative proof where the Galois space PG(3,n) in which \( \Omega \) is embedded is built up directly by using concepts and results on finite linear spaces.