Upper bounds on the uniform spreads of the sporadic simple groups

Abstract

\u200e\u200eA finite group $G$ has uniform spread $k$ if there exists a fixed conjugacy class $C$ of elements in $G$ with the property that\u200e \u200efor any $k$ nontrivial elements $s_1, s_2,\u200eldots\u200e,s_k$ in $G$ there exists $yin C$ such that $G = langle s_i,yrangle$ for $i=1, 2,\u200eldots,k$\u200e. \u200eFurther\u200e, \u200ethe exact uniform spread of $G$ is the largest $k$ such that $G$ has the uniform spread $k$\u200e. \u200eIn this paper we give upper bounds on the exact uniform spreads of thirteen sporadic simple groups\u200e.