The exact spread of M23 is 8064

Document Type: Research Paper

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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 $y\in G$ with the property that $\langle x_i,y\rangle=G$ for every $1\leq i\leq r$‎. ‎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$_{11}$‎.

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J.D. Bradley and P.E. Holmes (2007). Improved bounds for the spread of sporadic groups. LMS J Comput. Math.. 10, 132-140
J.D. Bradley and J. Moori (2007). On the exact spread of sporadic simple groups. Comm. Algebra. 35 (8), 2588-2599
J.L. Brenner and J. Wiegold (1975). Two-Generator Groups I. Michigan Math. J.. 22, 53-64
J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson (1985). An ATLAS of finite groups. Oxford University Press, Oxford,.
B.T. Fairbairn New upper bounds on the spreads of the sporadic simple groups. Comm. Algebra.
M.S. Ganief (1996). 2-Generations of the Sporadic Simple Groups. Ph.D thesis, University of Natal.
S. Ganief and J. Moori (2001). On the spread of the sporadic simple groups. Comm. Algebra. 29 (8), 3239-3255
A. Woldar (2007). The exact spread of the Mathieu group M11. J. Group Theory. 10, 167-171