A variation of Thompson's conjecture for the symmetric groups

Mahdi Abedei; Ali Iranmanesh; Farrokh Shirjian

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 3, page 743-755
  • ISSN: 0011-4642

Abstract

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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 ≅ 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 ≅ 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 ( Sym ( p + 1 ) ) , then G ≅ Sym ( p + 1 ) .

How to cite

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Abedei, Mahdi, Iranmanesh, Ali, and Shirjian, Farrokh. "A variation of Thompson's conjecture for the symmetric groups." Czechoslovak Mathematical Journal 70.3 (2020): 743-755. <http://eudml.org/doc/297006>.

@article{Abedei2020,
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 $G \cong \{\rm Sym\}(p+1)$.},
author = {Abedei, Mahdi, Iranmanesh, Ali, Shirjian, Farrokh},
journal = {Czechoslovak Mathematical Journal},
keywords = {Thompson's conjecture; conjugacy class size; symmetric groups; prime graph},
language = {eng},
number = {3},
pages = {743-755},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A variation of Thompson's conjecture for the symmetric groups},
url = {http://eudml.org/doc/297006},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Abedei, Mahdi
AU - Iranmanesh, Ali
AU - Shirjian, Farrokh
TI - A variation of Thompson's conjecture for the symmetric groups
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 3
SP - 743
EP - 755
AB - 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 $G \cong {\rm Sym}(p+1)$.
LA - eng
KW - Thompson's conjecture; conjugacy class size; symmetric groups; prime graph
UR - http://eudml.org/doc/297006
ER -

References

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