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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The classification of almost simple $ \tfrac{3}{2}$-transitive groups


Authors: John Bamberg, Michael Giudici, Martin W. Liebeck, Cheryl E. Praeger and Jan Saxl
Journal: Trans. Amer. Math. Soc. 365 (2013), 4257-4311
MSC (2010): Primary 20B05, 20B15
Published electronically: March 12, 2013
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Abstract: A finite transitive permutation group is said to be $ \frac {3}{2}$-transitive if all the nontrivial orbits of a point stabiliser have the same size greater than 1. Examples include the 2-transitive groups, Frobenius groups and several other less obvious ones. We prove that $ \frac {3}{2}$-transitive groups are either affine or almost simple, and classify the latter. One of the main steps in the proof is an arithmetic result on the subdegrees of groups of Lie type in characteristic $ p$: with some explicitly listed exceptions, every primitive action of such a group is either 2-transitive, or has a subdegree divisible by $ p$.


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Additional Information

John Bamberg
Affiliation: Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009
Email: John.Bamberg@uwa.edu.au

Michael Giudici
Affiliation: Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009
Email: Michael.Giudici@uwa.edu.au

Martin W. Liebeck
Affiliation: Department of Mathematics, Imperial College, London SW7 2BZ, England
Email: m.liebeck@imperial.ac.uk

Cheryl E. Praeger
Affiliation: Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009 – and – King Abdulaziz University, Jeddah, Saudi Arabia
Email: Cheryl.Praeger@uwa.edu.au

Jan Saxl
Affiliation: DPMMS, CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England
Email: saxl@dpmms.cam.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05758-3
PII: S 0002-9947(2013)05758-3
Received by editor(s): March 29, 2011
Received by editor(s) in revised form: November 15, 2011, and November 25, 2011
Published electronically: March 12, 2013
Additional Notes: This paper forms part of an Australian Research Council Discovery Project. The second author was supported by an Australian Research Fellowship while the fourth author was supported by a Federation Fellowship of the Australian Research Council.
Article copyright: © Copyright 2013 American Mathematical Society