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The exact spread of M23 IS 8064 | ||
| International Journal of Group Theory | ||
| مقاله 1، دوره 1، شماره 1، خرداد 2012، صفحه 1-2 اصل مقاله (240.8 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2012.466 | ||
| نویسنده | ||
| B. Fairbairn | ||
| چکیده | ||
| 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$_{11}$. | ||
| کلیدواژهها | ||
| Exact spread؛ sporadic group؛ finite simple group | ||
| مراجع | ||
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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
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