Geodesic
On recognition of alternating groups by prime graph
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 994-1010

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The prime graph $GK(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order $rs$. Let $Alt_n$ denote the alternating group of degree $n$. Assume that $p\geq13$ is a prime and $n$ is an integer such that $p\leq n\leq p+3$. We prove that if $G$ is a finite group such that $GK(G)=GK(Alt_n)$, then $G$ has a unique nonabelian composition factor, and this factor is isomorphic to $Alt_t$, where $p\leq t\leq p+3$.
Keywords: alternating group, prime graph, simple groups. 
@article{SEMR_2017_14_a28,
     author = {A. M. Staroletov},
     title = {On recognition of alternating groups by prime graph},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {994--1010},
     publisher = {mathdoc},
     volume = {14},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2017_14_a28/}
}
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A. M. Staroletov. On recognition of alternating groups by prime graph. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 994-1010. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a28/