Geodesic
Finite groups with $\mathbb{P}$-subnormal Schmidt subgroups
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 100-108
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A subgroup $H$ of a group $G$ is called $\mathbb{P}$-subnormal in $G$ whenever either $H = G$ or there is a chain of subgroups $$H = H_0 \subset H_1 \subset \ldots \subset H_n = G$$ such that $|H_i:H_{i-1}|$ is a prime for every $i = 1, 2,\ldots, n$. We study the structure of a finite group $G$ all of whose Schmidt subgroups are $\mathbb{P}$-subnormal. The obtained results complement the answer to Problem 18.30 in the Kourovka Notebook.
Keywords: finite group, $\mathbb{P}$-subnormal subgroup, Schmidt subgroup, saturated Fitting formation. 
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X. Yi; Zh. Xu; S. F. Kamornikov. Finite groups with $\mathbb{P}$-subnormal Schmidt subgroups. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 100-108. http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a7/

[1] Vasilev A.F., Vasileva T.I., Tyutyanov V.N., “O konechnykh gruppakh sverkhrazreshimogo tipa”, Sib. mat. zhurn., 51:6 (2010), 1270–1281 | MR | Zbl

[2] Vasilev A.F., Vasileva T.I., Tyutyanov V.N., “O proizvedeniyakh $\mathbb{P}$-subnormalnykh podgrupp v konechnykh gruppakh”, Sib. mat. zhurn., 53:1 (2012), 59–65 | MR

[3] Kniahina V.N., Monakhov V.S., “On supersolvability of finite groups with $\mathbb{P}$-subnormal subgroups”, Int. J. Group Theory, 2:4 (2013), 21–29 | DOI | MR | Zbl

[4] Monakhov V.S., “Konechnye gruppy s abnormalnymi i $\mathfrak{U}$-subnormalnymi podgruppami”, Sib. mat. zhurn., 57:2 (2016), 447–462 | DOI | MR | Zbl

[5] Ballester-Bolinches A., Li Y., Pedraza-Aguilera M.C., Su N., “On factorised finite groups”, Mediterr. J. Math., 17 (2020), 65 | DOI | MR | Zbl

[6] 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/

[7] Tyutyanov V.N., “Konechnye gruppy s $\mathbb{P}$-subnormalnymi podgruppami Shmidta”, Problemy fiziki, matematiki i tekhniki, 2015, no. 1(22), 88–91 | Zbl

[8] Kniahina V.N., Monakhov V.S., “Finite groups with $\mathbb{P}$-subnormal Sylow subgroups”, Ukr. Math. J., 72:10 (2021), 1571–1578 | DOI | MR | Zbl

[9] Berkovich Y.G., Janko Z., Groups of prime power order, v. 3, Walter de Gruyter, Berlin; NY, 2011, 668 pp. | DOI | MR | Zbl

[10] Ballester-Bolinches A., Kamornikov S.F., Pérez-Calabuig V., Tyutyanov V.N., “Finite groups with hereditarily $G$-permutable Schmidt subgroups”, Bull. Austral. Math. Soc., 2023, 1–7 | DOI | MR

[11] Vedernikov V.A., “Konechnye gruppy s subnormalnymi podgruppami Shmidta”, Algebra i logika, 46:6 (2007), 669–687 | MR | Zbl

[12] Doerk K., Hawkes T., Finite soluble groups, Walter de Gruyter, Berlin, NY, 1992, 891 pp. | DOI | MR

[13] Shmidt O.Yu., “Gruppy, vse podgruppy kotorykh spetsialnye”, Mat. sb., 31:3-4 (1924), 366–372

[14] Golfand Yu.A., “O gruppakh, vse podgruppy kotorykh spetsialnye”, Dokl. AN SSSR, 60:8 (1948), 1313–1315 | Zbl

[15] Monakhov V.S., “O podgruppakh Shmidta konechnykh grupp”, Voprosy algebry, 1998, no. 13, 153–171

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

[17] Knyagina V.N., Monakhov V.S., “O konechnykh gruppakh s nekotorymi subnormalnymi podgruppami Shmidta”, Sib. mat. zhurn., 45:6 (2004), 1316–1322 | MR | Zbl

[18] Monakhov V.S., “O konechnykh gruppakh s zadannym naborom podgrupp Shmidta”, Mat. zametki, 58:5 (1995), 717–722 | MR | Zbl