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Issue: Nov 2008

Volume 11, Number 6

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On the single-orbit conjecture for uncoverings-by-bases

Robert F. Bailey¹,

Peter J. Cameron²

¹ Robert F. Bailey, School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada. E-mail:

² Peter J. Cameron, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom. E-mail:

Citation Information. Journal of Group Theory. Volume 11, Issue 6, Pages 845–850, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: 10.1515/JGT.2008.053, Available online: 30 / 09 / 2008

Publication History: Received: 18/07/2007; revised: 03/12/2007; published online: 30 / 09 / 2008

Abstract

Let G be a permutation group acting on a finite set Ω. An uncovering-by-bases (or UBB) for G is a set of bases for G such that any r-subset of Ω is disjoint from at least one base in , where , for d the minimum degree of G. The single-orbit conjecture asserts that for any finite permutation group G, there exists a UBB for G contained in a single orbit of G on its irredundant bases. We prove a case of this conjecture, for when G is k-transitive and has a base of size k + 1. Furthermore, in the more restricted case when G is primitive and has a base of size 2, we show how to construct a UBB of minimum possible size.

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