Point regular automorphism groups of generalised quadrangles

John and I have just uploaded to the arxiv a copy of our recent paper, `Point regular automorphism groups of generalised quadrangles‘. We investigate the regular subgroups of some of the known generalised quadrangles. We demonstrate that the class of groups which can act as a point regular group of automorphisms of a generalised quadrangle is much wilder than previously thought.

A permutation group {G} on a set {\Omega} acts regularly on a set {\Omega} if it acts transitively on {\Omega} and only the identity of {G} fixes an element of {\Omega}. Studying regular automorphism groups of projective planes has received much attention over the years. Recently attention has turned to the study of groups acting regularly on generalised quadrangles.

Dina Ghinelli proved in 1992 that a Frobenius group or a group with a nontrivial centre cannot act regularly on the points of a generalised quadrangle of order {(s,s)}, where {s} is even. Stefaan De Winter and Koen Thas proved in 2006 that if a finite thick generalised quadrangle admits an abelian group of automorphisms acting regularly on its points, then it is the Payne derivation of a translation generalised quadrangle of even order. Satoshi Yoshiara proved that there are no generalised quadrangles of order {(s^2 , s)} admitting an automorphism group acting regularly on points.

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