Abstract
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.
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Notes
The original definition from [6] requires only the radicality of the class \(\mathfrak{X}\); however, as follows from the definition, it is natural to assume a priori that the class \(\mathfrak{X}\) is also closed under taking subgroups.
Formally, it was proved in [16] that the Baer–Suzuki width of the class of all groups of odd order is at most 2. However, it easily follows that \({\operatorname{BS}(\mathfrak{\mathfrak{G}_{\pi}})\leq 2}\) for any set \(\pi\) of odd primes. The latter was proved independently, although much later, in [17, Theorem 1].
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Funding
J. Guo and W. Guo were supported by the National Natural Science Foundation of China (project nos. 11961017 and 12171126). D. O. Revin and V. N. Tyutyanov were supported by the joint grant of the Russian Foundation for Basic Research (project no. 20-51-00007) and the Belarusian Republican Foundation for Fundamental Research (project no. F20R-291). D. O. Revin was also supported by the Program for Fundamental Research of the Russian Academy of Sciences (project no. FWNF-2022-0002).
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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 28, No. 2, pp. 96 - 105, 2022 https://doi.org/10.21538/0134-4889-2022-28-2-96-105.
Translated by E. Vasil’eva
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Guo, J., Guo, W., Revin, D.O. et al. On the Baer–Suzuki Width of Some Radical Classes. Proc. Steklov Inst. Math. 317 (Suppl 1), S90–S97 (2022). https://doi.org/10.1134/S0081543822030075
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DOI: https://doi.org/10.1134/S0081543822030075