the exact spread of m23 is 8064

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$_{11}$‎.