Squaring a conjugacy class and cosets of normal subgroups
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- by Robert Guralnick and Gabriel Navarro PDF
- Proc. Amer. Math. Soc. 144 (2016), 1939-1945 Request permission
Abstract:
Let $G$ be a finite group and let $K$ be the conjugacy class of $x \in G$. If $K^2$ is a conjugacy class of $G$, then $[x,G]$ is solvable. If the order of $x$ is a power of prime, then $[x,G]$ has a normal $p$-complement. We also prove some related results on the solvability of certain normal subgroups when a non-trivial coset has certain properties.References
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Additional Information
- Robert Guralnick
- Affiliation: Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089
- MR Author ID: 78455
- Email: guralnic@usc.edu
- Gabriel Navarro
- Affiliation: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain
- MR Author ID: 129760
- Email: gabriel.navarro@uv.es
- Received by editor(s): April 11, 2015
- Received by editor(s) in revised form: May 19, 2015, June 17, 2015, and June 21, 2015
- Published electronically: October 2, 2015
- Additional Notes: The first author gratefully acknowledges the support of the NSF grant DMS-1302886. The research of the second author was supported by the Prometeo/Generalitat Valenciana, Proyecto MTM2013-40464-P and FEDER funds. He would like to express his gratitude to the Mathematics Department of the University of Southern California, where part of the present work was completed, for its warm hospitality. Both authors thank M. Isaacs for helpful comments on the paper.
- Communicated by: Phan Huu Tiep
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1939-1945
- MSC (2010): Primary 20D06; Secondary 20D20
- DOI: https://doi.org/10.1090/proc/12874
- MathSciNet review: 3460157