## 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

Zbl 1318.20012
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