Czechoslovak Mathematical Journal, Vol. 68, No. 1, pp. 121-130, 2018
Recognition of some families of finite simple groups by order and set of orders of vanishing elements
Maryam Khatami, Azam Babai
Received July 7, 2016. First published January 12, 2018.
Abstract: 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.
Keywords: finite simple groups; vanishing element; vanishing prime graph
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Affiliations: Maryam Khatami, Department of Mathematics, University of Isfahan, HezarJerib Str., Isfahan 81746-73441, Iran, e-mail: m.khatami@sci.ui.ac.ir; Azam Babai, Department of Mathematics, University of Qom, Alghadir Blvd., Qom, P. O. Box 37185-3766, Iran, e-mail: a_babai@aut.ac.ir
