Normal coverings of linear groups

2012/06/19 by John R. Britnell, Britnell, John R., Attila Maroti +1
Mathematics · #20D50 (Primary) 20G40 (Secondary) #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20D50 #msc:20G40

paper · pdf · doi:10.48550/arxiv.1206.4279

18 pages

arxiv created 2012/06/19 · arxiv updated 2012/06/20

Abstract

For a non-cyclic finite group G, let γ(G) denote the smallest number of conjugacy classes of proper subgroups of G needed to cover G. Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently established that γ(S_n) and γ(A_n) are bounded above and below by linear functions of n. In this paper we show that if G is in the range \SL_n(q)≤ G≤ \GL_n(q) for n>2, then n/π² < γ(G) ≤ (n+1)/2. We give various alternative bounds, and derive explicit formulas for γ(G) in some cases.

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