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On Pyber's base size conjecture


Authors: Timothy C. Burness and Ákos Seress
Journal: Trans. Amer. Math. Soc. 367 (2015), 5633-5651
MSC (2010): Primary 20B15
DOI: https://doi.org/10.1090/S0002-9947-2015-06224-2
Published electronically: February 3, 2015
MathSciNet review: 3347185
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Abstract: Let $ G$ be a permutation group on a finite set $ \Omega $. A subset of $ \Omega $ is a base for $ G$ if its pointwise stabilizer in $ G$ is trivial. The base size of $ G$, denoted $ b(G)$, is the smallest size of a base. A well-known conjecture of Pyber from the early 1990s asserts that there exists an absolute constant $ c$ such that $ b(G) \leqslant c\log \vert G\vert/\log n$ for any primitive permutation group $ G$ of degree $ n$. Several special cases have been verified in recent years, including the almost simple and diagonal cases. In this paper, we prove Pyber's conjecture for all non-affine primitive groups.


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Additional Information

Timothy C. Burness
Affiliation: School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom
Address at time of publication: School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Email: t.burness@soton.ac.uk, t.burness@bristol.ac.uk

Ákos Seress
Affiliation: Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, Ohio 43210
Email: akos@math.ohio-state.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06224-2
Keywords: Finite permutation groups, primitive groups, base size, Pyber's conjecture
Received by editor(s): April 8, 2013
Received by editor(s) in revised form: June 19, 2013
Published electronically: February 3, 2015
Additional Notes: The first author was supported by EPSRC grant EP/I019545/1, and he thanks the Department of Mathematics at The Ohio State University for its generous hospitality
Ákos Seress passed away on February 13, 2013, shortly before this paper was submitted for publication
Article copyright: © Copyright 2015 American Mathematical Society