A variation of Thompson's conjecture for the symmetric groups

Abedei, Mahdi ; Iranmanesh, Ali ; Shirjian, Farrokh

Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 743-755.

Résumé

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 $G \cong {\rm Sym}(p+1)$.

Détail

DOI : 10.21136/CMJ.2020.0501-18

Classification : 20D08, 20D60
Keywords: Thompson's conjecture; conjugacy class size; symmetric groups; prime graph