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Base sizes for sporadic simple groups

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Abstract

Let G be a permutation group acting on a set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. We write b(G) for the minimal size of a base for G. We determine the precise value of b(G) for every primitive almost simple sporadic group G, with the exception of two cases involving the Baby Monster group. As a corollary, we deduce that b(G) ⩽ 7, with equality if and only if G is the Mathieu group M24 in its natural action on 24 points. This settles a conjecture of Cameron.

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Authors and Affiliations

  1. School of Mathematics, University of Southampton, Southampton, SO17 1BJ, UK

    Timothy C. Burness

  2. Department of Mathematics, University of Auckland, Auckland, New Zealand

    E. A. O’Brien

  3. School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK

    Robert A. Wilson

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  1. Timothy C. Burness
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  2. E. A. O’Brien
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  3. Robert A. Wilson
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Correspondence to Timothy C. Burness.

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Burness, T.C., O’Brien, E.A. & Wilson, R.A. Base sizes for sporadic simple groups. Isr. J. Math. 177, 307–333 (2010). https://doi.org/10.1007/s11856-010-0048-3

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  • DOI: https://doi.org/10.1007/s11856-010-0048-3

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