# Groups, Geometry, and Dynamics

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**Volume 2, Issue 1, 2008, pp. 85–120**

**DOI: 10.4171/GGD/32**

A commutator description of the solvable radical of a finite group

Nikolai Gordeev^{[1]}, Fritz Grunewald

^{[2]}, Boris Kunyavskii

^{[3]}and Eugene Plotkin

^{[4]}(1) Department of Mathematics, Herzen State Pedagogical University, 48, Moika Embankment, 191186, ST. PETERSBURG, RUSSIAN FEDERATION

(2) Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstrasse 1, 40225, DÜSSELDORF, GERMANY

(3) Department of Mathematics, Bar-Ilan University, 52900, RAMAT GAN, ISRAEL

(4) Department of Mathematics, Bar-Ilan University, 52900, RAMAT GAN, ISRAEL

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 `a`_{1}, `a`_{2}, … , `a`_{k} ∈ `G` the
subgroup generated by the elements `g`, `a`_{i}`g``a`_{i}^{−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
`b`_{1}, `b`_{2}, … , `b`_{ℓ} ∈ `G` the subgroup generated by the
commutators [`g`, `b`_{i}], `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.

*Keywords: *Finite group, solvable radical, simple group

Gordeev N, Grunewald F, Kunyavskii B, Plotkin E. A commutator description of the solvable radical of a finite group. *Groups Geom. Dyn.* 2 (2008), 85-120. doi: 10.4171/GGD/32