Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Czechoslovak Mathematical Journal

Maryam Khatami, Azam Babai¹ 

关键词: finite simple groups;    vanishing element;    vanishing prime graph;   

DOI  :  10.21136/CMJ.2018.0355-16

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

  Let $G$ be a finite group. An element $g\in G$ is called a vanishing element if there exists an irreducible complex character $\chi$ of $G$ such that $\chi(g)=0$. Denote by ${\rm Vo}(G)$ the set of orders of vanishing elements of $G$. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let $G$ be a finite group and $M$ a finite nonabelian simple group such that ${\rm Vo}(G)={\rm Vo}(M)$ and $|G|=|M|$. Then $G\cong M$. We answer in affirmative this conjecture for $M=Sz(q)$, where $q=2^{2n+1}$ and either $q-1$, $q-\sqrt{2q}+1$ or $q+\sqrt{2q}+1$ is a prime number, and $M=F_4(q)$, where $q=2^n$ and either $q^4+1$ or $q^4-q^2+1$ is a prime number.

【 授权许可】

Unknown