On the Baer–Suzuki Width of Some Radical Classes

Let $\sigma=\{\sigma_i\mid i\in I\}$ be a fixed partition of the set of all primes into pairwise disjoint nonempty subsets $\sigma_i$. A finite group is called $\sigma$-nilpotent if it has a normal $\sigma_i$-Hall subgroup for any $i\in I$. Any finite group possesses a $\sigma$-nilpotent radical, which is the largest normal $\sigma$-nilpotent subgroup. In this note, it is proved that there exists an integer $m=m(\sigma)$ such that the $\sigma$-nilpotent radical of any finite group coincides with the set of elements $x$ such that any $m$ conjugates of $x$ generate a $\sigma$-nilpotent subgroup. Other possible analogs of the classical Baer–Suzuki theorem are discussed.