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

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## Abstract

Let *G* be a finite group. An element *g* ∈ *G* is called a vanishing element if there exists an irreducible complex character χ of *G* such that χ(*g*)= 0. Denote by 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 Vo(*G*) = Vo(*M*) and |*G*| = |*M*|. Then *G* ≌ *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## MSC 2010

20C15 20D05## Preview

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© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018