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 with the property: if is a conjugacy class of a finite group such that every elements of generate a solvable subgroup, then generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take . We also present proofs that do not use the Classification Theorem. The most direct proof gives a value of . By lengthening one of our arguments slightly, we obtain a value of .

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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:
https://doi.org/10.1090/S0002-9939-09-10066-7

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