W. Guo, A.S. Kondrat’ev, N.V. Maslova, L. Miao. Finite groups whose maximal subgroups are solvable or have prime power indices ... P. 125-131

It is well known that all maximal subgroups of a finite solvable group are solvable and have prime power indices. However, the converse statement does not hold. Finite nonsolvable groups in which all local subgroups are solvable were studied by J. Thompson (1968). R. Guralnick (1983) described all the pairs $(G,H)$ such that $G$ is a finite nonabelian simple group and $H$ is a subgroup of prime power index in $G$. Several authors studied finite groups in which every subgroup of non-prime-power index (not necessarily maximal) is a group close to nilpotent. Weakening the conditions, E.N. Bazhanova (Demina) and N.V. Maslova (2014) considered the class $\mathfrak{J}_{\rm pr}$ of finite groups in which all nonsolvable maximal subgroups have prime power indices and, in particular, described possibilities for nonabelian composition factors of a nonsolvable group from the class $\mathfrak{J}_{\rm pr}$. In the present note, the authors continue the study of the normal structure of a nonsolvable group from $\mathfrak{J}_{\rm pr}$. It is proved that a group from $\mathfrak{J}_{\rm pr}$ contains at most one nonabelian chief factor and, for each positive integer $n$, there exists a group from $\mathfrak{J}_{\rm pr}$ such that the number of its nonabelian composition factors is at least $n$. Moreover, all almost simple groups from $\mathfrak{J}_{\rm pr}$ are determined.

Keywords:  finite group, maximal subgroup, prime power index, nonsolvable subgroup

Received April 23, 2020

Revised May 15, 2020

Accepted May 25, 2020

Funding Agency: This work was supported by a joint program of the Russian Foundation for Basic Research and the National Natural Science Foundation of China (project nos. 20-51-53013 and 12011530061), by the National Natural Science Foundation of China (project nos. 11771409 and 11871062), by the Natural Science Foundation of Jiangsu Province (project no. BK20181451), and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Wenbin Guo, Dr. Phys.-Math. Sci., School of Science, Hainan University, Haikou, Hainan, 570228 China; and School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026 China, e-mail: wbguo@ustc.edu.cn

Anatolii Semenovich Kondrat’ev, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: A.S.Kondratiev@imm.uran.ru

Natalia Vladimirovna Maslova, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University Yekaterinburg, 620083 Russia, e-mail: butterson@mail.ru

Long Miao, Ph. D., School of Mathematical Sciences, Yangzhou University, Yangzhou, 225002 China e-mail: lmiao@yzu.edu.cn

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Cite this article as: W. Guo, A.S. Kondrat’ev, N.V. Maslova, L. Miao. Finite groups whose maximal subgroups are solvable or have prime power indices. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 2, pp. 125–131.