# A commutator description of the solvable radical of a finite group

### Nikolai Gordeev

Herzen State Pedagogical University, St. Petersburg, Russian Federation### Fritz Grunewald

Heinrich-Heine-Universität, Düsseldorf, Germany### Boris Kunyavskii

Bar-Ilan University, Ramat Gan, Israel### Eugene Plotkin

Bar-Ilan University, Ramat Gan, Israel

## Abstract

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 $g,a_{i}ga_{i}$, $i=1, … ,k$, 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 $[g, bi]$, $i=1, … ,ℓ$, 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.

## Cite this article

Nikolai Gordeev, Fritz Grunewald, Boris Kunyavskii, Eugene Plotkin, A commutator description of the solvable radical of a finite group. Groups Geom. Dyn. 2 (2008), no. 1, pp. 85–120

DOI 10.4171/GGD/32