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Base sizes for simple groups and a conjecture of Cameron

Base sizes for simple groups and a conjecture of Cameron
Base sizes for simple groups and a conjecture of Cameron
Let G be a permutation group on a finite set ?. A base for G is a subset B C_ ? whose pointwise stabilizer in G is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ? if G is an almost simple group of exceptional Lie type and 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.
0024-6115
116-162
Burness, Timothy C.
a3b369f0-16f5-41e6-84d9-5a50f027bcd6
Liebeck, Martin W.
799394bb-c36c-4941-b91e-cf46a5971fa1
Shalev, Aner
afcf624b-1ba2-4a03-a0ce-af44343bfad4
Burness, Timothy C.
a3b369f0-16f5-41e6-84d9-5a50f027bcd6
Liebeck, Martin W.
799394bb-c36c-4941-b91e-cf46a5971fa1
Shalev, Aner
afcf624b-1ba2-4a03-a0ce-af44343bfad4

Burness, Timothy C., Liebeck, Martin W. and Shalev, Aner (2009) Base sizes for simple groups and a conjecture of Cameron. Proceedings of the London Mathematical Society, 98 (3), 116-162. (doi:10.1112/plms/pdn024).

Record type: Article

Abstract

Let G be a permutation group on a finite set ?. A base for G is a subset B C_ ? whose pointwise stabilizer in G is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ? if G is an almost simple group of exceptional Lie type and 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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Submitted date: 2 April 2007
Published date: 2009

Identifiers

Local EPrints ID: 69937
URI: https://eprints.soton.ac.uk/id/eprint/69937
ISSN: 0024-6115
PURE UUID: 70d6e00c-b57b-45cd-944a-f36d32b58232

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Date deposited: 05 Jan 2010
Last modified: 19 Jul 2017 00:05

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