Vol. 7, No. 9, 2013

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ISSN: 1944-7833 (e-only)
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Normal coverings of linear groups

John R. Britnell and Attila Maróti

Vol. 7 (2013), No. 9, 2085–2102
Abstract

For a noncyclic finite group G, let γ(G) denote the smallest number of conjugacy classes of proper subgroups of G needed to cover G. In this paper, we show that if G is in the range SLn(q) G GLn(q) for n > 2, then nπ2 < γ(G) (n + 1)2. This result complements recent work of Bubboloni, Praeger and Spiga on symmetric and alternating groups. We give various alternative bounds and derive explicit formulas for γ(G) in some cases.

Keywords
covering, normal covering, linear group, finite group
Mathematical Subject Classification 2010
Primary: 20D60
Secondary: 20G40
Milestones
Received: 28 July 2012
Revised: 1 November 2012
Accepted: 14 January 2013
Published: 18 December 2013
Authors
John R. Britnell
Department of Mathematics
Imperial College London
South Kensington Campus
London SW7 2AZ
United Kingdom
Attila Maróti
Hungarian Academy of Sciences
Alfréd Rényi Institute of Mathematics
Reáltanoda utca 13–15
H-1053, Budapest
Hungary