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Base sizes for S-actions of finite classical groups

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Abstract

Let G be a permutation group on a set Ω. A subset B of Ω is a base for G if the pointwise stabilizer of B in G is trivial; the base size of G is the minimal cardinality of a base for G, denoted by b(G). In this paper we calculate the base size of every primitive almost simple classical group with point stabilizer in Aschbacher’s collection S of irreducibly embedded almost simple subgroups. In this situation we also establish strong asymptotic results on the probability that randomly chosen subsets of Ω form a base for G. Indeed, with some specific exceptions, we show that almost all pairs of points in Ω are bases.

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Timothy C. Burness

  2. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Robert M. Guralnick

  3. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, UK

    Jan Saxl

Authors
  1. Timothy C. Burness
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  2. Robert M. Guralnick
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  3. Jan Saxl
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Correspondence to Timothy C. Burness.

Additional information

The authors would like to thank Dr. T. Breuer for his assistance with a computer calculation (see Lemma 5.5). They also thank Dr. C. Roney-Dougal for supplying the list of subgroups in S in the low dimensional classical groups (see Table 14), which is taken from the forthcoming book [7]. The first author was supported by EPSRC grant EP/I019545/1. The second author was partially supported by NSF grant DMS-1001962 and by Simons Foundation Fellowship 224965.

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Burness, T.C., Guralnick, R.M. & Saxl, J. Base sizes for S-actions of finite classical groups. Isr. J. Math. 199, 711–756 (2014). https://doi.org/10.1007/s11856-013-0059-y

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