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Nilpotency of normal subgroups having two $ G$-class sizes


Authors: Elena Alemany, Antonio Beltrán and María José Felipe
Journal: Proc. Amer. Math. Soc. 139 (2011), 2663-2669
MSC (2010): Primary 20E45, 20D15
DOI: https://doi.org/10.1090/S0002-9939-2010-10702-5
Published electronically: December 22, 2010
MathSciNet review: 2801605
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Abstract: Let $ G$ be a finite group. If $ N$ is a normal subgroup which has exactly two $ G$-conjugacy class sizes, then $ N$ is nilpotent. In particular, we show that $ N$ is abelian or is the product of a $ p$-group $ P$ by a central subgroup of $ G$. Furthermore, when $ P$ is not abelian, $ P/({\bf Z}(G)\cap P)$ has exponent $ p$.


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Additional Information

Elena Alemany
Affiliation: Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, 46022 Valencia, Spain
Email: ealemany@mat.upv.es

Antonio Beltrán
Affiliation: Departamento de Matemáticas, Universidad Jaume I, 12071 Castellón, Spain
Email: abeltran@mat.uji.es

María José Felipe
Affiliation: Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, 46022 Valencia, Spain
Email: mfelipe@mat.upv.es

DOI: https://doi.org/10.1090/S0002-9939-2010-10702-5
Received by editor(s): June 3, 2010
Received by editor(s) in revised form: July 14, 2010
Published electronically: December 22, 2010
Additional Notes: This work is part of the first author’s Ph.D. thesis and is partially supported by Proyecto MTM2007-68010-C03-03 and by Proyecto GV-2009-021
The second author is also supported by grant Fundació Caixa-Castelló P11B2008-09.
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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