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 $G$ h...
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| Format: | Article |
| Language: | English |
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University of Isfahan
2012-03-01
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| Series: | International Journal of Group Theory |
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| Online Access: | http://www.theoryofgroups.ir/?_action=showPDF&article=466&_ob=67971e04c1bbbbb3f887331dce0d6a2e&fileName=full_text.pdf |
| Summary: | 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$_{11}$. |
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| ISSN: | 2251-7650 2251-7669 |