A description of Baer-Suzuki type of the solvable radical of a finite group |
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Authors: | Nikolai Gordeev |
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Affiliation: | a Department of Mathematics, Herzen State Pedagogical University, 48 Moika Embankment, 191186, St.Petersburg, Russia b Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany c Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel |
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Abstract: | We obtain the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g∈G such that for any 3 elements a1,a2,a3∈G the subgroup generated by the elements , i=1,2,3, is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on {2,3}′-elements in the solvable radical of a finite group (which does not use the classification of finite simple groups). |
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Keywords: | 20F16 20D05 20D06 20D08 20D25 20G40 |
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