Overgroups of Cyclic Sylow Subgroups of Linear Groups |
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Authors: | John Bamberg Tim Penttila |
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Institution: | 1. School of Mathematics and Statistics , The University of Western Australia , Crawley, Western Australia, Australia bamberg@cage.ugent.be;3. School of Mathematics and Statistics , The University of Western Australia , Crawley, Western Australia, Australia |
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Abstract: | We use a theorem of Guralnick, Penttila, Praeger, and Saxl to classify the subgroups of the general linear group (of a finite dimensional vector space over a finite field) which are overgroups of a cyclic Sylow subgroup. In particular, our results provide the starting point for the classification of transitive m-systems; which include the transitive ovoids and spreads of finite polar spaces. We also use our results to prove a conjecture of Cameron and Liebler on irreducible collineation groups having equally many orbits on points and on lines. |
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Keywords: | Cameron–Liebler conjecture m-system Matrix group Ovoid Primitive prime divisor Spread |
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