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Base sizes for simple groups and a conjecture of Cameron
Author(s) -
Burness Timothy C.,
Liebeck Martin W.,
Shalev Aner
Publication year - 2009
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdn024
Subject(s) - mathematics , conjecture , base (topology) , combinatorics , simple (philosophy) , primitive permutation group , pointwise , simple group , permutation group , permutation (music) , group (periodic table) , set (abstract data type) , symmetric group , discrete mathematics , cyclic permutation , computer science , mathematical analysis , philosophy , epistemology , chemistry , physics , organic chemistry , acoustics , programming language
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.