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Abstract
The classification of flagtransitive generalized quadrangles is a longstanding open problem at the interface of finite geometry and permutation group theory. Given that all known flagtransitive generalized quadrangles are also pointprimitive (up to point–line duality), it is likewise natural to seek a classification of the pointprimitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on , the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that cannot have holomorph compound O’Nan–Scott type. Our arguments also pose purely grouptheoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.
Original language  English 

Pages (fromto)  87126 
Number of pages  40 
Journal  Nagoya Mathematical Journal 
Volume  234 
Early online date  14 Sept 2017 
DOIs  
Publication status  Published  Jun 2019 
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Dive into the research topics of 'Simple groups, product actions, and generalised quadrangles'. Together they form a unique fingerprint.Projects
 1 Finished

Finite geometry from an algebraic point of view
ARC Australian Research Council
1/01/12 → 30/06/17
Project: Research