Base sizes for simple groups and a conjecture of Cameron

Timothy C. Burness; Martin W. Liebeck; Aner Shalev

doi:10.1112/plms/pdn024

Base sizes for simple groups and a conjecture of Cameron

Timothy C. Burness, Martin W. Liebeck, Aner Shalev

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

56 Scopus citations

Abstract

Let G be a permutation group on a finite set Ω. A base for G is a subset B ⊆ Ω with pointwise stabilizer in G that is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ≤ 6 if G is an almost simple group of exceptional Lie type and Ω is a primitive faithful G-set. An important consequence of this result, when combined with other recent work, is that b(G) ≤ 7 for any almost simple group G in a non-standard action, proving a conjecture of Cameron. The proof is probabilistic and uses bounds on fixed point ratios.

Original language	English
Pages (from-to)	116-162
Number of pages	47
Journal	Proceedings of the London Mathematical Society
Volume	98
Issue number	1
DOIs	https://doi.org/10.1112/plms/pdn024
State	Published - Jan 2009

Bibliographical note

Funding Information:
The first author acknowledges the support of a Junior Research Fellowship from St John’s College, Oxford, and a Lady Davis Fellowship from The Hebrew University of Jerusalem.

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10.1112/plms/pdn024

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title = "Base sizes for simple groups and a conjecture of Cameron",

abstract = "Let G be a permutation group on a finite set Ω. A base for G is a subset B ⊆ Ω with pointwise stabilizer in G that is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ≤ 6 if G is an almost simple group of exceptional Lie type and Ω is a primitive faithful G-set. An important consequence of this result, when combined with other recent work, is that b(G) ≤ 7 for any almost simple group G in a non-standard action, proving a conjecture of Cameron. The proof is probabilistic and uses bounds on fixed point ratios.",

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Base sizes for simple groups and a conjecture of Cameron

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