×

Recognition of \(\mathrm{PSL}(2,2)^a)\) by the orders of vanishing elements. (English) Zbl 1401.20014

Summary: Here, we show that the simple groups \(\mathrm{PSL}(2,2)^a)\), \(a\geq2\), are characterized by the orders of vanishing elements.

MSC:

20C15 Ordinary representations and characters
20D06 Simple groups: alternating groups and groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20C33 Representations of finite groups of Lie type
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Conway J H, Curtis R T, Norton S P, Park R A and Wilson R A, Atlas of finite groups (1985) (Oxford: Clarendon Press) · Zbl 0568.20001
[2] Dolfi, S.; Pacifici, E.; Sanus, L.; Spiga, P., On the orders of zeros of irreducible characters, J. Algebra, 321, 345-352, (2009) · Zbl 1162.20005 · doi:10.1016/j.jalgebra.2008.10.004
[3] Dolfi, S.; Pacifici, E.; Sanus, L.; Spiga, P., On the vanishing prime graph of finite groups, J. London Math. Soc., 82, 167-183, (2010) · Zbl 1203.20024 · doi:10.1112/jlms/jdq021
[4] Dolfi, S.; Pacifici, E.; Sanus, L.; Spiga, P., On the vanishing prime graph of solvable groups, J. Group Theory, 13, 189-206, (2010) · Zbl 1196.20029 · doi:10.1515/jgt.2009.046
[5] Ghasemabadi, MF; Iranmanesh, A.; Mavadatpour, F., A new characterization of some finite simple groups, Sib. Math. J., 56, 78-82, (2015) · Zbl 1318.20012 · doi:10.1134/S0037446615010073
[6] Huppert B, Character Theory of Finite Groups (1998) (Berlin: de Gruyter) · Zbl 0932.20007 · doi:10.1515/9783110809237
[7] Huppert B and Blackburn N, Finite groups III (1982) (Berlin: Springer) · Zbl 0514.20002 · doi:10.1007/978-3-642-67997-1
[8] Isaacs I M, Character theory of finite groups (1976) (Cambridge: Academic Press) · Zbl 0337.20005
[9] Jiang Q H, Shao C G and Zhang J S, New characterization of PSL\((2,2^a)\) by character degree graph and order, submitted
[10] Khosravi, B.; Khosravi, B.; Khosravi, B.; Momen, Z., Recognition of the simple group PSL\((2, p^2)\) by character degree graph and order, Monatsh. Math., 178, 251-257, (2015) · Zbl 1325.20004 · doi:10.1007/s00605-014-0678-3
[11] Kondratev, AS, On prime graph components of finite simple groups, Mat. Sb., 180, 787-797, (1989)
[12] Manz, O.; Staszewski, R.; Willems, W., On the number of components of a graph related to character degrees, Proc. Am. Math. Soc., 103, 31-37, (1988) · Zbl 0645.20005 · doi:10.1090/S0002-9939-1988-0938639-1
[13] Shi, WJ, A characteristic property of \(J_1\) and PSL\((2, 2^n)\), Adv. Math. (China), 16, 397-401, (1987)
[14] Suzuki, M., Finite groups with nilpotent centralizers, Trans. Am. Math. Soc., 99, 425-470, (1961) · Zbl 0101.01604 · doi:10.1090/S0002-9947-1961-0131459-5
[15] Williama, JS, Prime graph components of finite groups, J. Algebra, 69, 487-513, (1981) · Zbl 0471.20013 · doi:10.1016/0021-8693(81)90218-0
[16] Zhang J S, Shen Z C and Shao C G, Recognition of some finite simple groups by the orders of vanishing elements, submitted
[17] Zhang, J. S.; Shao, C. G.; Shen, Z. C., A new characterization of Suzuki’s simple groups, J. Algebra Appl, 16, 6, (2017) · Zbl 1375.20005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.