The exact spread of M۲۳ is ۸۰۶۴

 Let $G$ be a finite group‎. ‎We say that $G$ has \emph{spread} r if for any set of distinct non-trivial elements of $G$ $X:=\{x_۱,\ldots‎, ‎x_r\}\subset G^{\#}$ there exists an element $y\in G$ with the property that $\langle x_i,y\rangle=G$ for every $۱\leq i\leq r$‎. ‎We say $G$ has \emph{exact spread} $r$ if $G$ has spread $r$ but not $r+۱$‎. ‎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 ۱۹۷۰s‎. ‎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$_{۲۳}$‎, ‎proving that it is equal to ۸۰۶۴‎. ‎The precise value of the exact spread of a sporadic simple group is known in only one other case‎ - ‎the Mathieu group M$_{۱۱}$‎.

کلیدواژه ها:

Exact spread ، sporadic group ، finite simple group