Geodesic
A new characterization of Mathieu groups
Archivum mathematicum, Tome 46 (2010) no. 1, pp. 13-23.

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Let $G$ be a finite group and $\operatorname{nse}(G)$ the set of numbers of elements with the same order in $G$. In this paper, we prove that a finite group $G$ is isomorphic to $M$, where $M$ is one of the Mathieu groups, if and only if the following hold: (1) $|G|=|M|$, (2) $\operatorname{nse}(G)=\operatorname{nse}(M)$.
Classification : 20D06, 20D60
Keywords: finite group; solvable group; order of element 
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     author = {Shao, Changguo and Jiang, Qinhui},
     title = {A new characterization of {Mathieu} groups},
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     publisher = {mathdoc},
     volume = {46},
     number = {1},
     year = {2010},
     mrnumber = {2644451},
     zbl = {1227.20007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2010__46_1_a1/}
}
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Shao, Changguo; Jiang, Qinhui. A new characterization of Mathieu groups. Archivum mathematicum, Tome 46 (2010) no. 1, pp. 13-23. http://geodesic.mathdoc.fr/item/ARM_2010__46_1_a1/