A solvable version of the Baer–Suzuki theorem
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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.References
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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
- MR Author ID: 890209
- Email: sguest@usc.edu
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 5909-5946
- MSC (2000): Primary 20F14, 20D10
- DOI: https://doi.org/10.1090/S0002-9947-2010-04932-3
- MathSciNet review: 2661502