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Recognition of some families of finite simple groups by order and set of orders of vanishing elements
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 1, pp. 121-130
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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.
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.
DOI : 10.21136/CMJ.2018.0355-16
Classification : 20C15, 20D05
Keywords: finite simple groups; vanishing element; vanishing prime graph 
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Khatami, Maryam; Babai, Azam. Recognition of some families of finite simple groups by order and set of orders of vanishing elements. Czechoslovak Mathematical Journal, Tome 68 (2018) no. 1, pp. 121-130. doi: 10.21136/CMJ.2018.0355-16

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