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On the maximum orders of elements of finite almost simple groups and primitive permutation groups


Authors: Simon Guest, Joy Morris, Cheryl E. Praeger and Pablo Spiga
Journal: Trans. Amer. Math. Soc. 367 (2015), 7665-7694
MSC (2010): Primary 20B15, 20H30
DOI: https://doi.org/10.1090/S0002-9947-2015-06293-X
Published electronically: March 23, 2015
MathSciNet review: 3391897
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Abstract: We determine upper bounds for the maximum order of an element of a finite almost simple group with socle $ T$ in terms of the minimum index $ m(T)$ of a maximal subgroup of $ T$: for $ T$ not an alternating group we prove that, with finitely many exceptions, the maximum element order is at most $ m(T)$. Moreover, apart from an explicit list of groups, the bound can be reduced to $ m(T)/4$. These results are applied to determine all primitive permutation groups on a set of size $ n$ that contain permutations of order greater than or equal to $ n/4$.


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Additional Information

Simon Guest
Affiliation: Centre for Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, Western Australia 6009, Australia
Address at time of publication: Peterborough Court, 133 Fleet Street, London EC4A 2BB, United Kingdom
Email: guest.simon@gmail.com

Joy Morris
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada
Email: joy.morris@uleth.ca

Cheryl E. Praeger
Affiliation: Centre for Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, Western Australia 6009, Australia; Department of Mathematics, King Abdulazziz University, Jeddah, Saudi Arabia
Email: Cheryl.Praeger@uwa.edu.au

Pablo Spiga
Affiliation: Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Via Cozzi 53, 20125 Milano, Italy
Email: pablo.spiga@unimib.it

DOI: https://doi.org/10.1090/S0002-9947-2015-06293-X
Keywords: Primitive permutation groups, conjugacy classes, cycle structure
Received by editor(s): January 22, 2013
Received by editor(s) in revised form: August 2, 2013
Published electronically: March 23, 2015
Additional Notes: The second author was supported in part by the National Science and Engineering Research Council of Canada
This research was supported in part by the Australian Research Council grants FF0776186 and DP130100106.
Article copyright: © Copyright 2015 by the authors

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