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Generalized Quadrangles and Transitive Pseudo‐Hyperovals
Author(s) -
Bamberg John,
Glasby S. P.,
Popiel Tomasz,
Praeger Cheryl E.
Publication year - 2016
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21411
Subject(s) - mathematics , combinatorics , transitive relation , linear subspace , abelian group , flag (linear algebra) , automorphism , automorphism group , projective space , dimension (graph theory) , pure mathematics , algebra over a field , projective test
A pseudo‐hyperoval of a projective spacePG ( 3 n − 1 , q ) , q even, is a set ofq n + 2 subspaces of dimension n − 1 such that any three span the whole space. We prove that a pseudo‐hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles Q that admit a point‐primitive, line‐transitive automorphism group with a point‐regular abelian normal subgroup. Specifically, we show that Q is flag‐transitive and isomorphic toT 2 * ( H ) , where H is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).