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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W.

Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus

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1435-4446
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Blocks of small defect in alternating groups and squares of Brauer character degrees

Xiaoyou Chen
/ James P. Cossey
/ Mark L. Lewis
/ Hung P. Tong-Viet
Published Online: 2017-07-18 | DOI: https://doi.org/10.1515/jgth-2017-0025

Abstract

Let p be a prime. We show that other than a few exceptions, alternating groups will have p-blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G has a normal Sylow p-subgroup P and $G/P$ is nilpotent if and only if $\phi {\left(1\right)}^{2}$ divides $|G:\mathrm{ker}\left(\phi \right)|$ for every irreducible Brauer character φ of G.

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Revised: 2016-06-23

Published Online: 2017-07-18

Published in Print: 2017-11-01

Funding Source: Henan University of Technology

Award identifier / Grant number: 2014JCYJ14

Award identifier / Grant number: 2016JJSB074

Award identifier / Grant number: 26510009

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11571129

The first author thanks the support of China Scholarship Council, Department of Mathematical Sciences of Kent State University for its hospitality, Funds of Henan University of Technology (2014JCYJ14, 2016JJSB074, 26510009), the Project of Department of Education of Henan Province (17A110004), the Projects of Zhengzhou Municipal Bureau of Science and Technology (20150249, 20140970), and the NSFC (11571129).

Citation Information: Journal of Group Theory, Volume 20, Issue 6, Pages 1155–1173, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883,

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