# Article

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Keywords:
finite simple groups; vanishing element; vanishing prime graph
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 ${\rm Vo}(G)$ 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.
References:
[1] Chen, G.: Further reflections on Thompson's conjecture. J. Algebra 218 (1999), 276-285. DOI 10.1006/jabr.1998.7839 | MR 1704687 | Zbl 0931.20020
[2] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A.: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon Press, Oxford (1985). MR 0827219 | Zbl 0568.20001
[3] Crescenzo, P.: A Diophantine equation which arises in the theory of finite groups. Adv. Math. 17 (1975), 25-29. DOI 10.1016/0001-8708(75)90083-3 | MR 0371812 | Zbl 0305.10016
[4] Dolfi, S., Pacifici, E., Sanus, L., Spiga, P.: On the vanishing prime graph of finite groups. J. Lond. Math. Soc., II. Ser. 82 (2010), 167-183. DOI 10.1112/jlms/jdq021 | MR 2669646 | Zbl 1203.20024
[5] Dolfi, S., Pacifici, E., Sanus, L., Spiga, P.: On the vanishing prime graph of solvable groups. J. Group Theory 13 (2010), 189-206. DOI 10.1515/JGT.2009.046 | MR 2607575 | Zbl 1196.20029
[6] Ghasemabadi, M. F., Iranmanesh, A., Ahanjideh, M.: A new characterization of some families of finite simple groups. Rend. Semin. Mat. Univ. Padova 137 (2017), 57-74. DOI 10.4171/RSMUP/137-3 | MR 3652868 | Zbl 06735308
[7] Ghasemabadi, M. F., Iranmanesh, A., Mavadatpur, F.: A new characterization of some finite simple groups. Sib. Math. J. 56 (2015), 78-82 English. Russian original translation from Sib. Math. Zh. 56 2015 94-99. DOI 10.1134/S0037446615010073 | MR 3407941 | Zbl 1318.20012
[8] Isaacs, I. M.: Character Theory of Finite Groups. Pure and Applied Mathematics 69, Academic Press, New York (1976). DOI 10.1515/9783110809237 | MR 0460423 | Zbl 0337.20005
[9] James, G., Liebeck, M.: Representations and Characters of Groups. Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge (1993). DOI 10.1017/CBO9780511814532 | MR 1237401 | Zbl 0792.20006
[10] Shi, H., Chen, G. Y.: Relation between $B_p(3)$ and $C_p(3)$ with their order components where $p$ is an odd prime. J. Appl. Math. Inform. 27 (2009), 653-659.
[11] Vasil'ev, A. V., Vdovin, E. P.: An adjacency criterion for the prime graph of a finite simple group. Algebra Logic 44 (2005), 381-406 English. Russian original translation from Algebra Logika 44 2005 682-725. DOI 10.1007/s10469-005-0037-5 | MR 2213302 | Zbl 1104.20018
[12] Williams, J. S.: Prime graph components of finite groups. J. Algebra 69 (1981), 487-513. DOI 10.1016/0021-8693(81)90218-0 | MR 0617092 | Zbl 0471.20013
[13] Zhang, J., Li, Z., Shao, C.: Finite groups whose irreducible characters vanish only on elements of prime power order. Int. Electron. J. Algebra 9 (2011), 114-123. MR 2753762 | Zbl 1259.20010
[14] Zsigmondy, K.: Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3 (1892), 265-284 German \99999JFM99999 24.0176.02. DOI 10.1007/BF01692444 | MR 1546236

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