Base sizes for simple groups and a conjecture of Cameron

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

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

30 Citations (Scopus)

Abstract

Let G be a permutation group on a finite set S. A base for G is a subset B of S 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) is at most 6 if G is an almost simple group of exceptional Lie type and S is a primitive faithful G-set. An important consequence of this result, when combined with other recent work, is that b(G) is at most 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 languageEnglish
Pages (from-to)116-162
Number of pages47
JournalProceedings of the London Mathematical Society
Volume98
Issue number1
Early online date24 Jun 2008
DOIs
Publication statusPublished - Jan 2009

Keywords

  • PERMUTATION-GROUPS
  • FINITE CLASSICAL-GROUPS
  • MAXIMAL-SUBGROUPS
  • CHEVALLEY-GROUPS
  • LIE TYPE
  • EXCEPTIONAL ALGEBRAIC-GROUPS
  • FIXED-POINT RATIOS
  • CHARACTER SHEAVES
  • GREEN-FUNCTIONS
  • REDUCTIVE GROUPS

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