Characterizations of the solvable radical

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We prove that there exists a constant k with the property: if C is a conjugacy class of a finite group G such that every k elements of C generate a solvable subgroup, then 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.


Original languageEnglish
Pages (from-to)1161-1170
Number of pages10
JournalProceedings of the American Mathematical Society
Issue number04
Publication statusPublished - 1 Apr 2010


  • Solvable radical, generation by conjugates