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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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 
@article{TIMM_2022_28_2_a7,
     author = {J. Guo and W. Guo and D. O. Revin and V. N. Tyutyanov},
     title = {On the {Baer{\textendash}Suzuki} {Width} of {Some} {Radical} {Classes}},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {96--105},
     year = {2022},
     volume = {28},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2022_28_2_a7/}
}
TY  - JOUR
AU  - J. Guo
AU  - W. Guo
AU  - D. O. Revin
AU  - V. N. Tyutyanov
TI  - On the Baer–Suzuki Width of Some Radical Classes
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2022
SP  - 96
EP  - 105
VL  - 28
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMM_2022_28_2_a7/
LA  - ru
ID  - TIMM_2022_28_2_a7
ER  - 
%0 Journal Article
%A J. Guo
%A W. Guo
%A D. O. Revin
%A V. N. Tyutyanov
%T On the Baer–Suzuki Width of Some Radical Classes
%J Trudy Instituta matematiki i mehaniki
%D 2022
%P 96-105
%V 28
%N 2
%U http://geodesic.mathdoc.fr/item/TIMM_2022_28_2_a7/
%G ru
%F TIMM_2022_28_2_a7
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/

[1] Shemetkov L.A., Formatsii konechnykh grupp, Nauka, M., 1978, 271 pp. | MR

[2] Skiba A.N., “On $\sigma$-subnormal and $\sigma$-permutable subgroups of finite groups”, J. Algebra, 436 (2015), 1–16 | DOI | MR | Zbl

[3] Skiba A.N., “On some results in the theory of finite partially soluble groups”, Comm. Math. Stat., 4:3 (2016) | DOI | MR

[4] Baer R., “Engelsche Elemente Noetherscher Gruppen”, Math. Ann., 133:3 (1957), 256–270 | DOI | MR | Zbl

[5] Suzuki M., “Finite groups in which the centralizer of any element of order $2$ is $2$-closed”, Ann. Math., 82:1 (1965), 191–212 | DOI | MR | Zbl

[6] Gordeev N., Grunewald F., Kunyavskii B., Plotkin E., “A description of Baer–Suzuki type of the solvable radical of a finite group”, J. Pure Appl. Algebra, 213:2 (2009), 250–258 | DOI | MR | Zbl

[7] Alperin J., Lyons R., “On conjugacy classes of $p$-elements”, J. Algebra, 19:2 (1971), 536–537 | DOI | MR | Zbl

[8] Gorenstein D., Finite groups, 2nd ed., Chelsea P. C., NY, 1980, 519 pp. | MR | Zbl

[9] Gordeev N., Grunewald F., Kunyavskii B., Plotkin E., “Baer–Suzuki theorem for the solvable radical of a finite group”, C. R. Acad. Sci. Paris, Ser. I, 347:5–6 (2009), 217–222 | DOI | MR | Zbl

[10] Gordeev N., Grunewald F., Kunyavskii B., Plotkin E., “From Thompson to Baer–Suzuki: a sharp characterization of the solvable radical”, J. Algebra, 323:10 (2010), 2888–2904 | DOI | MR | Zbl

[11] Flavell P., Guest S., Guralnick R., “Characterizations of the solvable radical”, Proc. Amer. Math. Soc., 138:4 (2010), 1161–1170 | DOI | MR | Zbl

[12] Guest S., “A solvable version of the Baer–Suzuki theorem”, Trans. Amer. Math. Soc., 362:11 (2010), 5909–5946 | DOI | MR | Zbl

[13] Yang N., Revin D.O., Vdovin E.P., “Baer–Suzuki theorem for the $\pi$-radical”, Isr. J. Math., 245:1 (2021), 173–207 | DOI | MR | Zbl

[14] Yan N., U Chzh. Revin D.O., “O tochnoi teoreme Bera–Suzuki dlya $\pi$-radikala: sporadicheskie gruppy”, Sib. mat. zhurn., 63:2 (2022), 465–473 | DOI | Zbl

[15] Yan N., U Chzh., Revin D.O., Vdovin E.P., O tochnoi teoreme Bera–Suzuki dlya $\pi$-radikala, [e-resource](sdano v Mat. sb.), 2021, 36 pp., arXiv: 2105.02442

[16] Tyutyanov V.N., “Kriterii neprostoty dlya konechnoi gruppy”, Izv. Gomel. gos. un-ta. im. F. Skoriny. Voprosy algebry, 16:3 (2000), 125–137 | Zbl

[17] Revin D.O., “O $\pi$-teoremakh Bera–Sudzuki”, Sib. mat. zhurn., 52:2 (2011), 430–440 | MR | Zbl

[18] Wielandt H., “Zusammengesetzte Gruppen: Hölder Programm heute”, The Santa Cruz conf. on finite groups (Santa Cruz, 1979), Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, RI, 1980, 161–173 | DOI | MR

[19] The Kourovka notebook. Unsolved problems in group theory, 20th ed., eds. V.D. Mazurov, E.I. Khukhro, Inst. Math. SO RAN Publ., Novosibirsk, 2022, 269 pp. URL: https://kourovkanotebookorg.files.wordpress.com/2022/02/20tkt-3.pdf

[20] Isaacs I.M., Character theory of finite groups, Pure Appl. Math., 359, Acad. Press, NY, 1976, 303 pp. | MR