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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Characterizations of the solvable radical


Authors: Paul Flavell, Simon Guest and Robert Guralnick
Journal: Proc. Amer. Math. Soc. 138 (2010), 1161-1170
MSC (2010): Primary 20F14, 20D10
Published electronically: December 2, 2009
MathSciNet review: 2578510
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Abstract: We prove that there exists a constant $ k$ with the property: if $ \mathcal{C}$ is a conjugacy class of a finite group $ G$ such that every $ k$ elements of $ \mathcal{C}$ generate a solvable subgroup, then $ \mathcal{C}$ generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take $ k=4$. We also present proofs that do not use the Classification Theorem. The most direct proof gives a value of $ k=10$. By lengthening one of our arguments slightly, we obtain a value of $ k=7$.


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Additional Information

Paul Flavell
Affiliation: School of Mathematics, University of Birmingham, Birmingham B15 2TT, United Kingdom
Email: P.J.Flavell@bham.ac.uk

Simon Guest
Affiliation: Department of Mathematics, Baylor University, One Bear Place #97328, Waco, Texas 76798-7328
Email: Simon_Guest@baylor.edu

Robert Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
Email: guralnic@usc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10066-7
PII: S 0002-9939(09)10066-7
Keywords: Solvable radical, generation by conjugates
Received by editor(s): August 28, 2008
Published electronically: December 2, 2009
Additional Notes: The second and third authors were partially supported by NSF grant DMS 0653873.
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2009 American Mathematical Society