Czechoslovak Mathematical Journal, first online, pp. 1-13


A variation of Thompson's conjecture for the symmetric groups

Mahdi Abedei, Ali Iranmanesh, Farrokh Shirjian

Received November 9, 2018.   Published online January 29, 2020.

Abstract:  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 $G \cong{\rm Sym}(p+1)$.
Keywords:  Thompson's conjecture; conjugacy class size; symmetric groups; prime graph
Classification MSC:  20D08, 20D60
DOI:  10.21136/CMJ.2020.0501-18

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Affiliations:   Mahdi Abedei, Ali Iranmanesh (corresponding author), Farrokh Shirjian, Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-137, Tehran, Iran, e-mail: mahdi_abedi@modares.ac.ir, iranmanesh@modares.ac.ir, farrokh.shirjian@modares.ac.ir


 
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