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Squaring a conjugacy class and cosets of normal subgroups

Authors: Robert Guralnick and Gabriel Navarro
Journal: Proc. Amer. Math. Soc. 144 (2016), 1939-1945
MSC (2010): Primary 20D06; Secondary 20D20
Published electronically: October 2, 2015
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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.

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Robert Guralnick
Affiliation: Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089

Gabriel Navarro
Affiliation: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain

Keywords: Conjugacy class, coset of normal subgroup
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
Article copyright: © Copyright 2015 American Mathematical Society