Geodesic
On the Baer--Suzuki Width of Some Radical Classes
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 2, pp. 96-105

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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.
Keywords: Baer–Suzuki width, $\sigma$-nilpotent group, complete class of groups.
Mots-clés : $\sigma$-solvable group 
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     title = {On the {Baer--Suzuki} {Width} of {Some} {Radical} {Classes}},
     journal = {Trudy Instituta matematiki i mehaniki},
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J. Guo; W. Guo; D. O. Revin; V. N. Tyutyanov. On the Baer--Suzuki Width of Some Radical Classes. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 2, pp. 96-105. http://geodesic.mathdoc.fr/item/TIMM_2022_28_2_a7/