Normal coverings of linear groups

Britnell, John; Maróti, Attila

doi:10.2140/ant.2013.7.2085

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Normal coverings of linear groups

John R. Britnell and Attila Maróti

Vol. 7 (2013), No. 9, 2085–2102

DOI: 10.2140/ant.2013.7.2085

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 ${SL}_{n} (q) \leq G \leq {GL}_{n} (q)$ for $n > 2$ , then $n ∕ π^{2} < γ (G) \leq (n + 1) ∕ 2$ . 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

Vol. 7, No. 9, 2013