Even Character Degrees and Ito–Michler Theorem

Abstract

Let

{\text{Irr}}_2(G)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{Irr}_2(G)$$\end{document}

be the set of linear and even-degree irreducible characters of a finite group G. In this paper, we prove that G has a normal Sylow 2-subgroup if

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sum \limits _{\chi \in \textrm{Irr}_2(G)} \chi (1)^m/\sum \limits _{\chi \in \textrm{Irr}_2(G)} \chi (1)^{m-1} < (1+2^{m-1})/(1+2^{m-2})$$\end{document}

for a positive integer m, which is the generalization of several recent results concerning the well-known Ito–Michler theorem.

Keywords

Character degrees / Sylow subgroups / 20C15

References

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[2] Isaacs IM. Character Theory of Finite Groups. 1976, New York, Academic Press

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[4] Marinelli S, Tiep PH. Zeros of real irreducible characters of finite groups. Algebra Number Theory. 2013, 3: 567-593.

[5] Michler GO. Brauer’s conjectures and the classification of finite simple groups. Lect. Notes Math.. 1986, 1178: 129-142.

[6] Pan HF, Hung NN, Dong SQ. Even character degrees and normal Sylow 2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2$$\end{document}-subgroups. J. Group Theory. 2021, 24: 195-205.

Funding

National Natural Science Foundation of China(12201236)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature