Planar Singer groups with even order multiplier groups

Summary

Abstract

We completely determine the subgroups, which also are subplanes, of a Singer group of planar order 81. We prove that each subgroup of a Singer group is invariant under the involution of the multiplier group, except possibly if the Singer group is non abelian of planar order 16. If the subgroup is a subplane of non square order, then this subplane is centralized by the involution of the multiplier group. We study v(n) = v(x)v(y)v(z) from a geometrical point of view, where n is the order of a projective plane and v(r) = r² + r + 1 for any r.

Introduction

A Singer group of a projective plane is a collineation group acting regularly on the points of the plane. In 1938, Singer proved that a finite Desarguesian plane admits a cyclic Singer group. On the other hand, in 1964, Karzel proved that a plane admitting an infinite cyclic Singer group is not Desarguesian. Projective planes and Singer groups in this article are of finite cardinalities. An automorphism of a Singer group is a multiplier if it is also a collineation when we identify the points of the plane with the elements of the group. The set of all multipliers is called the multiplier group of the Singer group.