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
Received December 5, 2023
Revised January 8, 2024
Accepted January 15, 2024
Funding Agency: The research of the first author was supported by the National Natural Science Foundation of China (grant no. 12371021). The research of the third author was supported by the Belarusian Republican Foundation for Fundamental Research (project no. Φ23PHΦ-237).
Xiaolan Yi, Zhejiang Sci-Tech University, Hangzhou, P. R. China, e-mail: yixiaolan2005@126.com
Zhuyan Xu, Zhejiang Sci-Tech University, Hangzhou, P. R. China, e-mail: xuzhuyan2022@163.com
Sergei Fedorovich Kamornikov, Dr. Phys.-Math. Sci., Prof., F. Skorina Gomel State University, Gomel, 246028 Republic of Belarus, e-mail: sfkamornikov@mail.ru
REFERENCES
1. Vasil’ev A.F., Vasil’eva T.I., Tyutyanov V.N. On the finite groups of supersoluble type. Sib. Math. J., 2010, vol. 51, no. 6, pp. 1004–1012. doi:10.1007/s11202-010-0099-z
2. Vasil’ev A.F., Tyutyanov V.N., Vasil’eva T.I. On the products of $\mathbb{P}$-subnormal subgroups of finite groups. Sib. Math. J., 2012, vol. 53, no. 1, pp. 47–54. doi:10.1134/S0037446612010041
3. Kniahina V.N., Monakhov V.S. On supersolvability of finite groups with $\mathbb{P}$-subnormal subgroups. Int. J. Group Theory, 2013, vol. 2, no. 4, pp. 21–29. doi:10.22108/IJGT.2013.2835
4. Monakhov V.S. Finite groups with abnormal and $\mathfrak{U}$-subnormal subgroups. Sib. Math. J., 2016, vol. 57, no. 2, pp. 352–363. doi:10.1134/S0037446616020178
5. Ballester-Bolinches A., Li Y., Pedraza-Aguilera M.C., Su N. On factorised finite groups. Mediterr. J. Math., 2020, vol. 17, article 65. doi:10.1007/s00009-020-1500-1
6. The Kourovka notebook. Unsolved problems in group theory, 20th ed., eds. V.D. Mazurov, E.I. Khukhro, Novosibirsk, Inst. Math. SO RAN Publ., 2022, 269 p. Available at: https://kourovka-notebook.org/.
7. Tyutyanov V.N. Finite groups with $\mathbb{P}$-subnormal Schmidt subgroups. Problemy Fiziki, Matematiki i Tekhniki, 2015, no. 1, pp. 88–91 (in Russian).
8. Kniahina V.N., Monakhov V.S. Finite groups with $\mathbb{P}$-subnormal Sylow subgroups. Ukr. Math. J., 2021, vol. 72, no. 10, pp. 1571–1578. doi: 10.37863/umzh.v72i10.2264
9. Berkovich Y.G., Janko Z. Groups of prime power order. Volume 3. Berlin, NY, Walter de Gruyter, 2011, 668 p. doi: 10.1515/9783110254488
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, pp. 1–7. doi: 10.1017/s0004972723000771
11. Vedernikov V.A. Finite groups with subnormal Schmidt subgroups. Algebra and Logic, 2007, vol. 46, no. 6, pp. 363–372. doi:10.1007/s10469-007-0036-9
12. Doerk K., Hawkes T. Finite soluble groups. Berlin, NY, Walter de Gruyter, 1992, 891 p. doi: 10.1515/9783110870138
13. Schmidt O.Yu. Groups whose all subgroups are special, Mat. Sbornik, 1924, vol. 31, no. 3–4, pp. 366–372 (in Russian).
14. Gol’fand Yu.A. On groups all of whose subgroups are special. Dokl. Akad. Nauk SSSR, 1948, vol. 60, no. 8, pp. 1313–1315.
15. Monakhov V.S. The Schmidt subgroups of finite groups. Vopr. Algebry, 1998, vol. 13, pp. 153–171 (in Russian).
16. Shemetkov L.A. Formatsii konechnykh grupp [Formations of finite groups]. Moskow, Nauka Publ., 1978, 271 p.
17. Knyagina V.N., Monakhov V.S. Finite groups with subnormal Schmidt subgroups. Sib. Math. J., 2004, vol. 45, no. 6, pp. 1075–1079. doi: 10.1023/B:SIMJ.0000048922.59466.20
18. Monakhov V.S. Finite groups with a given set of Schmidt subgroups. Math. Notes, 1995, vol. 58, no. 5, pp. 1183–1186. doi: 10.1007/BF02305002
Cite this article as: X. Yi, Zh. Xu, S.F. Kamornikov. Finite groups with $\mathbb{P}$-subnormal Schmidt subgroups. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 100–108. Proceedings of the Steklov Institute of Mathematics, 2024, Vol. 325, Suppl. 1, pp. S231–S238.
