Geodesic
A commutator description of the solvable radical of a finite group
Nikolai Gordeev1 ; Fritz Grunewald2 ; Boris Kunyavskii3 ; Eugene Plotkin3
1Herzen State Pedagogical University, St. Petersburg, Russian Federation
2Heinrich-Heine-Universität, Düsseldorf, Germany
3Bar-Ilan University, Ramat Gan, Israel
Groups, geometry, and dynamics, Tome 2 (2008) no. 1, pp. 85-120

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We are looking for the smallest integer k > 1 providing 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 k elements a1​,a2​, ...,ak​∈G, the subgroup generated by the elements g,ai​gai−1​, i=1, ... ,k, is solvable. We consider a similar problem of finding the smallest integer l> 1 with the property that R(G) coincides with the collection of all g∈G such that for any l elements b1​,b2​,... ,bl​∈G, the subgroup generated by the commutators [g, bi], i=1, ... ,l, is solvable. Conjecturally, k=l=3. We prove that both k and l are at most 7. In particular, this means that a finite group G is solvable if and only if every 8 conjugate elements of G generate a solvable subgroup.
DOI : 10.4171/ggd/32
Classification : 20-XX, 00-XX
Mots-clés : Finite group, solvable radical, simple group

Nikolai Gordeev  1   ; Fritz Grunewald  2   ; Boris Kunyavskii  3   ; Eugene Plotkin  3

1 Herzen State Pedagogical University, St. Petersburg, Russian Federation
2 Heinrich-Heine-Universität, Düsseldorf, Germany
3 Bar-Ilan University, Ramat Gan, Israel 
Nikolai Gordeev; Fritz Grunewald; Boris Kunyavskii; Eugene Plotkin. A commutator description of the solvable radical of a finite group. Groups, geometry, and dynamics, Tome 2 (2008) no. 1, pp. 85-120. doi: 10.4171/ggd/32
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