Let be a finite group, and let be a nonnegative integer. We say that has uniform spread if there exists a fixed conjugacy class in with the property that for any nontrivial elements in there exists such that for all . Further, the exact uniform spread of , denoted by , is the largest such that has the uniform spread property. By a theorem of Breuer, Guralnick, and Kantor, for every finite simple group . Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if is almost simple, then (unless ), and we determine precisely when tends to infinity as tends to infinity.
"On the uniform spread of almost simple linear groups." Nagoya Math. J. 209 35 - 109, March 2013. https://doi.org/10.1215/00277630-1959460