Nilpotency of normal subgroups having two $G$-class sizes
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- by Elena Alemany, Antonio Beltrán and María José Felipe PDF
- Proc. Amer. Math. Soc. 139 (2011), 2663-2669 Request permission
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, $P/(\textbf {Z}(G)\cap P)$ has exponent $p$.References
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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
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2801605