Fixed points of projectivities of prime order

Abstract

This work begins with a review of the classical results for fixed points of projectivities in a projective plane over a general commutative field. The second section of this work features all the material necessary to prove the main result, which is presented in Theorem 2.8. It is shown that, in a finite projective plane of order q, there exists a projectivity g̃ of prime order p > 3 if and only if p divides exactly one of the integers q − 1, q, q + 1, q ² + q + 1. Theorem 2.8 establishes a correspondence between the possible structures of points fixed by g̃, as presented in Theorem 1.3, and the integer that is divisible by p. The special case of p = 2 is handled in Sect. 2.1, where it is shown that every involution is a harmonic homology for q odd and an elation for q even. The special case of p = 3 is handled in Sect. 2.2, and Theorem 2.8 is adapted for p = 3 and presented as Theorem 2.15. An application of Theorems 2.8 and 2.15 is determining the sizes of (n, r)-arcs that are stabilized by projectivities of prime order p in the finite projective plane of order q; in Sect. 3, this application is presented in Propositions 3.2 and 3.3.