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Journal of the London Mathematical Society Advance Access originally published online on June 18, 2007
Journal of the London Mathematical Society 2007 75(3):545-562; doi:10.1112/jlms/jdm033
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© 2007 London Mathematical Society

On base sizes for actions of finite classical groups

Timothy C. Burness

Institute of Mathematics
Hebrew University of Jerusalem
Jerusalem 91904
Israel

Let G be a finite almost simple classical group and let {Omega} be a faithful primitive non-standard G-set. A subset of {Omega} is a base for G if its pointwise stabilizer in G is trivial. Let b(G) be the minimal size of a base for G. A well-known conjecture of Cameron and Kantor asserts that there exists an absolute constant c such that b(G) ≤ slant c for all such groups G, and the existence of such an undetermined constant has been established by Liebeck and Shalev. In this paper we prove that either b(G) ≤ 4, or G = U6(2) · 2, G{omega} = U4(3) · 22 and b(G) = 5. The proof is probabilistic, using bounds on fixed point ratios.

burness{at}math.huji.ac.il

2000 Mathematics Subject Classification 20B15 (primary), 20P05 (secondary).

Received June 20, 2006; revised October 31, 2006; published online June 18, 2007.


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