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On the uniform spread of almost simple linear groups

Published online by Cambridge University Press:  11 January 2016

Timothy C. Burness  and
Simon Guest
Timothy C. Burness
Affiliation:
School of Mathematics University of Southampton, Southampton SO17 1BJ, UK, t.burness@soton.ac.uk
Simon Guest
Affiliation:
School of Mathematics University of Southampton, Southampton SO17 1BJ, UK, s.d.guest@soton.ac.uk
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Abstract

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Let G be a finite group, and let k be a nonnegative integer. We say that G has uniform spread k if there exists a fixed conjugacy class C in G with the property that for any k nontrivial elements x1,…,xk in G there exists y ∊ C such that G = ‹xi,y› for all i. Further, the exact uniform spread of G, denoted by u(G), is the largest k such that G has the uniform spread k property. By a theorem of Breuer, Guralnick, and Kantor, u(G) ≥ 2 for every finite simple group G. Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if G = ‹PSLn (q),g› is almost simple, then u(G) ≥ 2 (unless G ≅ S6), and we determine precisely when u(G) tends to infinity as |G| tends to infinity.

Keywords

20D0620E2820F0520P05
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 209 , March 2013 , pp. 35 - 109
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

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