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A solvable version of the Baer-Suzuki theorem

Author: Simon Guest
Journal: Trans. Amer. Math. Soc. 362 (2010), 5909-5946
MSC (2000): Primary 20F14, 20D10
Published electronically: June 2, 2010
MathSciNet review: 2661502
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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $ G$ is a finite group and $ x \in G$ has prime order $ p \ge 5$. Then $ x$ is contained in the solvable radical of $ G$, $ O_{\infty}(G)$, if (and only if) $ \langle x,x^g \rangle$ is solvable for all $ g \in G$. If $ G$ is an almost simple group and $ x \in G$ has prime order $ p \ge 5$, then this implies that there exists $ g \in G$ such that $ \langle x,x^g \rangle$ is not solvable. In fact, this is also true when $ p=3$ with very few exceptions, which are described explicitly.

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Additional Information

Simon Guest
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089–2532
Address at time of publication: Department of Mathematics, Baylor University, One Bear Place, #97328, Waco, Texas 76798

Received by editor(s): January 25, 2008
Received by editor(s) in revised form: September 14, 2008
Published electronically: June 2, 2010
Additional Notes: The author was partially supported by the NSF grant DMS 0653873
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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