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Finite primitive groups and regular orbits of group elements

Authors: Simon Guest and Pablo Spiga
Journal: Trans. Amer. Math. Soc. 369 (2017), 997-1024
MSC (2010): Primary 20B15, 20H30
Published electronically: April 15, 2016
MathSciNet review: 3572262
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Abstract: We prove that if $ G$ is a finite primitive permutation group and if $ g$ is an element of $ G$, then either $ g$ has a cycle of length equal to its order, or for some $ r$, $ m$ and $ k$, the group $ G \leq \mathrm {Sym}(m) \mathrm {wr}\mathrm {Sym}(r)$ preserves the product structure of $ r$ direct copies of the natural action of $ \mathrm {Sym}(m)$ on $ k$-sets. This gives an answer to a question of Siemons and Zalesski and a solution to a conjecture of Giudici, Praeger and the second author.

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Additional Information

Simon Guest
Affiliation: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom

Pablo Spiga
Affiliation: Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Via Cozzi 55 Milano, MI 20125, Italy

Keywords: Cycle lengths, element orders, primitive groups
Received by editor(s): June 5, 2014
Received by editor(s) in revised form: December 29, 2014, and February 2, 2015
Published electronically: April 15, 2016
Additional Notes: The second author is the corresponding author
Article copyright: © Copyright 2016 American Mathematical Society

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