X. Yi, Zh. Xu, S.F. Kamornikov. Finite groups with $\mathbb{P}$-subnormal Schmidt subgroups ... P. 100-108

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

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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.