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Irreducible characters of even degree and normal Sylow 2-subgroups

Published online by Cambridge University Press:  15 July 2016

NGUYEN NGOC HUNG  and
PHAM HUU TIEP
NGUYEN NGOC HUNG
Affiliation:
Department of Mathematics, The University of Akron, Akron, Ohio 44325, U.S.A. e-mail: [email protected]
PHAM HUU TIEP
Affiliation:
Department of Mathematics, University of Arizona, Tucson, Arizona 85721, U.S.A. e-mail: [email protected]
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Abstract

The classical Itô-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group G is coprime to a given prime p, then G has a normal Sylow p-subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of G is less than 4/3 then G has a normal Sylow 2-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These results improve on several earlier results concerning the Itô-Michler theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 2 , March 2017 , pp. 353 - 365
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A. Atlas of Finite Groups (Clarendon Press, Oxford, 1985).Google Scholar
[2] Cossey, J. P. and Nguyen, H. N. Controlling composition factors of a finite group by its character degree ratio. J. Algebra 403 (2014), 185–200.Google Scholar
[3] Cossey, J. P., Halasi, Z., Maróti, A. and Nguyen, H. N.. On a conjecture of Gluck. Math. Z. 279 (2015), 1067–1080.Google Scholar
[4] Dolfi, S., Navarro, G. and Tiep, Pham Huu. Primes dividing the degrees of the real characters. Math. Z. 259 (2008), 755–774.CrossRefGoogle Scholar
[5] Dolfi, S., Pacifici, E., Sanus, L. and Spiga, P.. On the orders of zeros of irreducible characters. J. Algebra 321 (2009), 345–352.Google Scholar
[6] Feit, W. Extending Steinberg characters. Linear algebraic groups and their representations. Contemp. Math. 153 (1993), 1–9.CrossRefGoogle Scholar
[7] Gluck, D. The largest irreducible character degree of a finite group. Canad. J. Math. 37 (1985), 442–451.Google Scholar
[8] Hung, N. N. Characters of p'-degree and Thompson's character degree theorem. Rev. Mat. Iberoam., to appear. http://arxiv.org/abs/1506.06450 Google Scholar
[9] Isaacs, I. M. ‘Character Theory of Finite Groups’ (AMS Chelsea Publishing, Providence, Rhode Island, 2006).Google Scholar
[10] Isaacs, I. M. The number of generators of a linear p-group. Canad. J. Math. 24 (1972), 851–858.CrossRefGoogle Scholar
[11] Isaacs, I. M. and Navarro, G. Groups whose real irreducible characters have degree coprime to p . J. Algebra 356 (2012), 195–206.Google Scholar
[12] Isaacs, I. M., Loukaki, M. and Moretó, A. The average degree of an irreducible character of a finite group. Israel J. Math. 197 (2013), 55–67.Google Scholar
[13] Itô, N. Some studies on group characters. Nagoya Math. J. 2 (1951), 17–28.Google Scholar
[14] Lewis, M. Ł. and Nguyen, H. N.. The character degree ratio and composition factors of a finite group. Monatsh. Math. 180 (2016), 289–294.CrossRefGoogle Scholar
[15] Manz, O. On the modular version of Ito's theorem on character degrees for groups of odd order. Nagoya Math. J. 105 (1987), 121–128.CrossRefGoogle Scholar
[16] Manz, O. and Wolf, T. R. Brauer characters of q'-degree in p-solvable groups. J. Algebra 115 (1988), 75–91.CrossRefGoogle Scholar
[17] Marinelli, S. and Tiep, Pham Huu. Zeros of real irreducible characters of finite groups. Algebra Number Theory 7 (2013), 567–593.Google Scholar
[18] Michler, G. O. Brauer's conjectures and the classification of finite simple groups. Lecture Notes in Math. 1178 (Springer, Berlin, 1986), 129–142.Google Scholar
[19] Michler, G. O. A finite simple group of Lie type has p-blocks with different defects, p ≠ 2. J. Algebra 104 (1986), 220–230.Google Scholar
[20] Moretó, A. and Nguyen, H. N. On the average character degree of finite groups. Bull. Lond. Math. Soc. 46 (2014), 454–462.Google Scholar
[21] Navarro, G. Characters and Blocks of Finite Groups. London Mathematical Society Lecture Note Series 250 (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
[22] Navarro, G. and Tiep, Pham Huu. Degrees of rational characters of finite groups. Adv. Math. 224 (2010), 1121–1142.Google Scholar
[23] Navarro, G. and Tiep, Pham Huu. Degrees and p-rational characters. Bull. Lond. Math. Soc. 44 (2012), 1246–1250.Google Scholar
[24] Rasala, R. On the minimal degrees of characters of $\textup{\textsf{S}}_{n}$ . J. Algebra 45 (1977), 132–181.Google Scholar
[25] Seitz, G. and Zalesskii, A. E. On the minimal degrees of projective representations of the finite Chevalley groups, II. J. Algebra 158 (1993), 233–243.Google Scholar
[26] Tiep, Pham Huu. Real ordinary characters and real Brauer characters. Trans. Amer. Math. Soc. 367 (2015), 1273–1312.Google Scholar