The exact spread of M23 is 8064

Abstract

Let $G$ be a finite group‎. ‎We say that $G$ has emph{spread} r if for any set of distinct non-trivial elements of $G$ $X:={x_1,ldots‎, ‎x_r}subset G^{#}$ there exists an element $yin G$ with the property that $langle x_i,yrangle=G$ for every $1leq ileq r$‎. ‎We say $G$ has emph{exact spread} $r$ if $G$ has spread $r$ but not $r+1$‎. ‎The spreads of finite simple groups and their decorations have been much-studied since the concept was first introduced by Brenner and Wiegold in the mid 1970s‎. ‎Despite this‎, ‎the exact spread of very few finite groups‎, ‎and in particular of the finite simple groups and their decorations‎, ‎is known‎. ‎Here we calculate the exact spread of the sporadic simple Mathieu group M$_{23}$‎, ‎proving that it is equal to 8064‎. ‎The precise value of the exact spread of a sporadic simple group is known in only one other case‎ - ‎the Mathieu group M$_{11}$‎.