Geodesic
A variation of Thompson's conjecture for the symmetric groups
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 743-755

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Let $G$ be a finite group and let $N(G)$ denote the set of conjugacy class sizes of $G$. Thompson's conjecture states that if $G$ is a centerless group and $S$ is a non-abelian simple group satisfying $N(G)=N(S)$, then $G\cong S$. In this paper, we investigate a variation of this conjecture for some symmetric groups under a weaker assumption. In particular, it is shown that $G\cong {\rm Sym}(p+1)$ if and only if $|G|=(p+1)!$ and $G$ has a special conjugacy class of size $(p + 1)!/p$, where $p>5$ is a prime number. Consequently, if $G$ is a centerless group with $N(G)=N({\rm Sym}(p+1))$, then $G \cong {\rm Sym}(p+1)$.
DOI : 10.21136/CMJ.2020.0501-18
Classification : 20D08, 20D60
Keywords: Thompson's conjecture; conjugacy class size; symmetric groups; prime graph 
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     title = {A variation of {Thompson's} conjecture for the symmetric groups},
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Abedei, Mahdi; Iranmanesh, Ali; Shirjian, Farrokh. A variation of Thompson's conjecture for the symmetric groups. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 743-755. doi: 10.21136/CMJ.2020.0501-18

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