A commutator description of the solvable radical of a finite group

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 $g\in G$ such that for any $k$ elements $a_1,a_2,\dots ,a_k\in G$, the subgroup generated by the elements $g,a_{i}g{a}_i^{-1}$, $i=1,\dots ,k$, is solvable. We consider a similar problem of finding the smallest integer $\ell >1$ with the property that $R(G)$ coincides with the collection of all $g\in G$ such that for any $\ell$ elements $b_1,b_2,\dots ,b_{\ell }\in G$, the subgroup generated by the commutators $[g,bi]$, $i=1,\dots ,\ell$, is solvable. Conjecturally, $k=\ell =3$. We prove that both $k$ and $\ell$ 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.