Nagoya Mathematical Journal

On the uniform spread of almost simple linear groups

Timothy C. Burness and Simon Guest

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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 yC 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 GS6), and we determine precisely when u(G) tends to infinity as |G| tends to infinity.

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Nagoya Math. J., Volume 209 (2013), 35-109.

First available in Project Euclid: 27 February 2013

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Zentralblatt MATH identifier

Primary: 20D06: Simple groups: alternating groups and groups of Lie type [See also 20Gxx]
Secondary: 20E28: Maximal subgroups 20F05: Generators, relations, and presentations 20P05: Probabilistic methods in group theory [See also 60Bxx]


Burness, Timothy C.; Guest, Simon. On the uniform spread of almost simple linear groups. Nagoya Math. J. 209 (2013), 35--109. doi:10.1215/00277630-1959460.

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