Thompson-like characterizations of the solvable radical

Robert Guralnick, Boris Kunyavskiǐ, Eugene Plotkin, Aner Shalev*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

35 Scopus citations

Abstract

We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x ∈ G the subgroup of G generated by x and y is solvable. This confirms a conjecture of Flavell. We present analogues of this result for finite-dimensional Lie algebras and some classes of infinite groups. We also consider a similar problem for pairs of elements.

Original languageAmerican English
Pages (from-to)363-375
Number of pages13
JournalJournal of Algebra
Volume300
Issue number1 SPEC. ISS.
DOIs
StatePublished - 1 Jun 2006

Bibliographical note

Funding Information:
✩ Guralnick was partially supported by the National Science Foundation Grant DMS 0140578. Kunyavski˘ı and Plotkin were partially supported by the Ministry of Absorption (Israel), the Israel Science Foundation founded by the Israel Academy of Sciences, Center of Excellence Program, the Minerva Foundation through the Emmy Noether Research Institute of Mathematics, and the EU networks HPRN-CT-2002-00287 and INTAS 00-566. Shalev was partially supported by the Israel Science Foundation and the Bi-National Science Foundation United States–Israel. * Corresponding author. E-mail addresses: guralnic@usc.edu (R. Guralnick), kunyav@macs.biu.ac.il (B. Kunyavski˘ı), plotkin@macs.biu.ac.il (E. Plotkin), shalev@math.huji.ac.il (A. Shalev).

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