Developments in finite Phan theory

An automorphism of a generalized quadrangle is called domestic if it maps no chamber, which is here an incident point-line pair, to an opposite chamber. We call it point-domestic if it maps no point to an opposite one and line-domestic if it maps no line to an opposite one. It is clear that a duality in a generalized quadrangle is always point-domestic and line-domestic.In this paper, we classify all domestic automorphisms of generalized quadrangles. Besides three exceptional cases occurring in the small quadrangles with orders (2, 2), (2, 4) and (3, 5), all domestic collineations are either point-domestic or line-domestic. Up to duality, they fall into one of three classes: either they are central collineations, or they fix an ovoid, or they fix a large full subquadrangle. Remarkably, the three exceptional domestic collineatons in the small quadrangles mentioned above all have order 4.

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The Combinatorics of Automorphisms and Opposition in Generalised Polygons

Parkinson

Temmermans

Maldeghem

2015

Ann. Comb.

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We investigate the combinatorial interplay between automorphisms and opposition in (primarily finite) generalised polygons. We provide restrictions on the fixed element structures of automorphisms of a generalised polygon mapping no chamber to an opposite chamber. Furthermore, we give a complete classification of automorphisms of finite generalised polygons which map at least one point and at least one line to an opposite, but map no chamber to an opposite chamber. Finally, we show that no automorphism of a finite thick generalised polygon maps all chambers to opposite chambers, except possibly in the case of generalised quadrangles with coprime parameters.

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Developments in finite Phan theory

Cited by 14 publications

References 98 publications

Characterizations of Trialities of Type $$\mathrm {I}_{\mathsf {id}}$$ in Buildings of Type $$\mathsf {D}_4$$

Characterizations of Trialities of Type $$\mathrm {I}_{\mathsf {id}}$$ in Buildings of Type $$\mathsf {D}_4$$

Domesticity in Generalized Quadrangles

The Combinatorics of Automorphisms and Opposition in Generalised Polygons

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