On generalized quadrangles with a point regular group of automorphisms

2017/10/24 by Swartz, Eric
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.09019

Abstract

A generalized quadrangle is a point-line incidence geometry such that any two points lie on at most one line and, given a line ℓ and a point P not incident with ℓ, there is a unique point of ℓ collinear with P. We study the structure of groups acting regularly on the point set of a generalized quadrangle. In particular, we provide a characterization of the generalized quadrangles with a group of automorphisms acting regularly on both the point set and the line set and show that such a thick generalized quadrangle does not admit a polarity. Moreover, we prove that a group G acting regularly on the point set of a generalized quadrangle of order (u², u³) or (s,s), where s is odd and s+1 is coprime to 3, cannot have any nonabelian minimal normal subgroups.

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