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  • JOHN BAMBERG (a1), TOMASZ POPIEL (a2) (a3) and CHERYL E. PRAEGER (a2) (a4)


The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$ , 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 $G$ cannot have holomorph compound O’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.



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John Bamberg is supported by the Australian Research Council Future Fellowship FT120100036. Tomasz Popiel and Cheryl E. Praeger are supported by the Australian Research Council Discovery Grant DP140100416.



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  • JOHN BAMBERG (a1), TOMASZ POPIEL (a2) (a3) and CHERYL E. PRAEGER (a2) (a4)


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