The structure of the central units of a commutative semifield plane

Summary

Abstract

Let π^l∞ be a semifield plane of order q^N, with middle nucleus GF(q). Relative to any fixed natural autotopism triangle, every fixed affine point I (not on the triangle) determines up to isomorphism a unique coordinatising semifield D_I. I is called a central unit if D_I is commutative. We determine the geometric distribution of the central units of π^l∞ and hence show that the plane has precisely (q^N – l)(q – 1) central units.

Introduction

Let π^l∞ be a semifield plane with an autotopism triangle OXY: where XY = l_∞ is the translation axis, and OY is a shears axis, with O ∈ π^l∞. Now each choice of a “unit point” I, off the chosen autotopism triangle (assumed fixed from now on) determines uniquely up to isomorphism a semifield D_I that coordinatises π^l∞. We shall call I a central unit relative to the chosen frame if D_I is a commutative semifield. By a criterion of Ganley [2] [theorem 3] the finite semifield planes admitting central units are precisely the finite translation planes that admit orthogonal polarities. However, no geometric characterisation of the set of central units in a given semifield plane has ever been recorded. The purpose of this note is to provide a geometric description of the distribution of the central units of a given commutative semifield plane.