Geodesic
On groups with Frobenius–Engel elements
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 213-222
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A number of properties of periodic and mixed groups with Frobenius–Engel elements are found (Lemmas in Sect. 2 and Theorem 1). The results obtained are used to describe mixed and periodic groups with finite elements saturated with finite Frobenius groups. It is proved that a binary finite group saturated with finite Frobenius groups is a Frobenius group with locally finite complement (Theorem 2). Theorem 3 establishes that in a saturated Frobenius group of a primitive binary finite group $G$ without involutions the characteristic subgroup $\Omega_1(G)$ generated by all elements of prime orders from $G$ is a periodic Frobenius group with kernel $F$ and locally cyclic complement $H$. Moreover, any maximal periodic subgroup $T$ of $G$ is a Frobenius group with kernel $F$ and complement $T\cap N_G(H)$. A number of examples of periodic non-locally finite and mixed groups satisfying Theorem 3 are given.
Keywords: Frobenius groups, finite elements, Engel elements, Frobenius elements, Frobenius–Engel elements
Mots-clés : saturation. 
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A. I. Sozutov. On groups with Frobenius–Engel elements. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 213-222. http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a15/

[1] Kurosh A.G., Teoriya grupp, 3-e izd., Nauka, Moskva, 1967, 648 pp. | MR

[2] Sozutov A.I., “O gruppakh, nasyschennykh konechnymi gruppami Frobeniusa”, Mat. zametki, 109:2 (2021), 264–275 | DOI | Zbl

[3] Popov A.M., Sozutov A.I., Shunkov V.P., Gruppy s sistemami frobeniusovykh podgrupp, IPTs KGTU, Krasnoyarsk, 2004, 211 pp.

[4] Durakov B.E., Sozutov A.I., “On periodic groups saturated with finite Frobenius groups”, Bulletin of Irkutsk State University. Ser. Mathematics, 35, 73–86 | DOI | MR | Zbl

[5] Durakov B.E., Sozutov A.I., “O gruppakh s involyutsiyami, nasyschennykh konechnymi gruppami Frobeniusa”, Sib. mat. zhurn., 63:6 (2022), 1256–1265 | DOI

[6] Shlepkin A.K., “O nekotorykh periodicheskikh gruppakh, nasyschennykh konechnymi prostymi gruppami”, Mat. tr., 1:1 (1998), 129–138 | MR | Zbl

[7] Maslova N.V, Shlepkin A. A., “O gruppakh Shunkova, nasyschennykh pochti prostymi gruppami”, Algebra i logika, 62:1 (2023), 93–101 | DOI

[8] Kukharev A.V., Shlepkin A.A., “Lokalno konechnye gruppy, nasyschennye pryamym proizvedeniem dvukh konechnykh grupp diedra”, Izv. Irkutskogo gos. un-ta. Ser. Matematika, 44 (2023), 71–81 | DOI | MR

[9] Shlepkin A.A., Sabodakh I.V., “O dvukh svoistvakh gruppy Shunkova”, Izv. Irkutskogo gos. un-ta. Ser. Matematika, 35 (2021), 103–119 | DOI | MR | Zbl

[10] Lytkina D.V., Mazurov V.D., “O periodicheskikh gruppakh, nasyschennykh konechnymi prostymi simplekticheskimi gruppami razmernosti 6 nad polyami nechetnykh kharakteristik”, Sib. mat. zhurn., 63:6 (2022), 1308–1312 | DOI

[11] Go V. B., Lytkina D. V., Mazurov V. D., “O periodicheskikh gruppakh, nasyschennykh konechnymi prostymi gruppami $L4(q)$”, Algebra i logika, 60:6 (2021), 549–556 | DOI

[12] Lytkina D.V., Mazurov V.D., “Lokalnaya konechnost periodicheskoi gruppy, nasyschennoi konechnymi prostymi ortogonalnymi gruppami nechetnoi razmernosti”, Sib. mat. zhurn., 62:3 (2021), 572–578 | DOI

[13] 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://kourovka-notebook.org/

[14] Starostin A.I., “O gruppakh Frobeniusa”, Ukr. mat. zhurn., 23:5 (1971), 629–639 | MR | Zbl