Base sizes for simple groups and a conjecture of Cameron

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

Research output: Contribution to journalArticlepeer-review

56 Scopus citations


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 languageAmerican English
Pages (from-to)116-162
Number of pages47
JournalProceedings of the London Mathematical Society
Issue number1
StatePublished - 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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