Geodesic
Recognition of alternating groups of degrees~$r+1$ and $r+2$ for prime~$r$ and the group
Algebra i logika, Tome 39 (2000) no. 6, pp. 648-661.

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It is proved that a finite group whose element order set is the same as that of an alternating group $A_n$ of degree $n=r+1$ or $r+2$ for prime $r>5$ or $n=16$ is isomorphic to $A_n$. 
@article{AL_2000_39_6_a1,
     author = {A. V. Zavarnitsin},
     title = {Recognition of alternating groups of degrees~$r+1$ and $r+2$ for prime~$r$ and the group},
     journal = {Algebra i logika},
     pages = {648--661},
     publisher = {mathdoc},
     volume = {39},
     number = {6},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2000_39_6_a1/}
}
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A. V. Zavarnitsin. Recognition of alternating groups of degrees~$r+1$ and $r+2$ for prime~$r$ and the group. Algebra i logika, Tome 39 (2000) no. 6, pp. 648-661. http://geodesic.mathdoc.fr/item/AL_2000_39_6_a1/