The classification of almost simple $\tfrac {3}{2}$-transitive groups
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- by John Bamberg, Michael Giudici, Martin W. Liebeck, Cheryl E. Praeger and Jan Saxl PDF
- Trans. Amer. Math. Soc. 365 (2013), 4257-4311 Request permission
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 $p$.References
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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
- MR Author ID: 655176
- ORCID: 0000-0001-5412-4656
- Email: Michael.Giudici@uwa.edu.au
- Martin W. Liebeck
- Affiliation: Department of Mathematics, Imperial College, London SW7 2BZ, England
- MR Author ID: 113845
- ORCID: 0000-0002-3284-9899
- 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
- MR Author ID: 141715
- ORCID: 0000-0002-0881-7336
- 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
- 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.
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 4257-4311
- MSC (2010): Primary 20B05, 20B15
- DOI: https://doi.org/10.1090/S0002-9947-2013-05758-3
- MathSciNet review: 3055696