Upper bounds on the uniform spreads of the sporadic simple groups

Abstract

‎‎A finite group $G$ has uniform spread $k$ if there exists a fixed conjugacy class $C$ of elements in $G$ with the property that‎ ‎for any $k$ nontrivial elements $s_1, s_2,‎ldots‎,s_k$ in $G$ there exists $yin C$ such that $G = langle s_i,yrangle$ for $i=1, 2,‎ldots,k$‎. ‎Further‎, ‎the exact uniform spread of $G$ is the largest $k$ such that $G$ has the uniform spread $k$‎. ‎In this paper we give upper bounds on the exact uniform spreads of thirteen sporadic simple groups‎.