Characterizations of the solvable radical
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- by Paul Flavell, Simon Guest and Robert Guralnick PDF
- Proc. Amer. Math. Soc. 138 (2010), 1161-1170 Request permission
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 $k=7$.References
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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
- MR Author ID: 78455
- Email: guralnic@usc.edu
- 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
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 1161-1170
- MSC (2010): Primary 20F14, 20D10
- DOI: https://doi.org/10.1090/S0002-9939-09-10066-7
- MathSciNet review: 2578510