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Recognition of some families of finite simple groups by order and set of orders of vanishing elements. (English) Zbl 1458.20009

Summary: 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 \(\text{Vo}(G)\) the set of orders of vanishing elements of \(G\). M. F. Ghasemabadi et al. [Sib. Math. J. 56, No. 1, 78–82 (2015; Zbl 1318.20012); translation from Sib. Mat. Zh. 56, No. 1, 94–99 (2015)], in their paper presented the following conjecture: Let \(G\) be a finite group and \(M\) a finite nonabelian simple group such that \(\text{Vo}(G)=\text{Vo}(M)\) 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.

MSC:

20C15 Ordinary representations and characters
20D05 Finite simple groups and their classification
20D60 Arithmetic and combinatorial problems involving abstract finite groups

Citations:

Zbl 1318.20012
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