Nagoya Mathematical Journal

On the uniform spread of almost simple linear groups

Abstract

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 $x_{1},\ldots,x_{k}$ in $G$ there exists $y\in C$ such that $G=\langle x_{i},y\rangle$ 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, $u(G)\ge2$ for every finite simple group $G$. Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if $G=\langle \operatorname {PSL}_{n}(q),g\rangle$ is almost simple, then $u(G)\ge2$ (unless $G\cong S_{6}$), and we determine precisely when $u(G)$ tends to infinity as $|G|$ tends to infinity.

Article information

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

Dates
First available in Project Euclid: 27 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1361976372

Digital Object Identifier
doi:10.1215/00277630-1959460

Mathematical Reviews number (MathSciNet)
MR3032138

Zentralblatt MATH identifier
1271.20012

Citation

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. https://projecteuclid.org/euclid.nmj/1361976372

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