MSC: 20D10, 20D20
DOI: 10.21538/0134-4889-2025-31-4-290-299
This work was supported by General Program of Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant # 2024D01C200), NSFC (Grant # 12371018), Fundamental Research Funds for the Central Universities (Grant # B240201093/2013), NSFC-RFBR (Grant # 12011530061), and Yili Normal University General Natural Science Project for Enhancing Comprehensive Discipline Strength (Grant # 22XKZY19).
In this paper, we extend the formation $\hat{\mathfrak{J}_{pr}}$, which is generated by the class $\mathfrak{J}_{pr}$ originally introduced by Demina and Maslova. The class $\mathfrak{J}_{pr}$ consists of finite groups in which every non-solvable maximal subgroup has a primary index. Building upon this framework, we introduce and study two generalized formations, denoted by $\hat{\mathfrak{J}}$ and $\hat{\mathfrak{J}_{p}}$, which are obtained by involving minimal non-solvable maximal subgroups and applying a localization approach to maximal subgroups. We establish new sufficient conditions under which a finite group belongs to these formations. In addition, we give examples of non-solvable groups to illustrate the distinctions between the class $\mathfrak{J}_{pr}$ and its generalizations.
Keywords: formation, non-solvable group, second maximal subgroup, the core of subgroup
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Received May 22, 2025
Revised August 25, 2025
Accepted September 8, 2025
Funding Agency: This work was supported by General Program of Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant # 2024D01C200), NSFC (Grant # 12371018), Fundamental Research Funds for the Central Universities (Grant # B240201093/2013), NSFC-RFBR (Grant # 12011530061), and Yili Normal University General Natural Science Project for Enhancing Comprehensive Discipline Strength (Grant # 22XKZY19).
Wenxia Zhou, School of Mathematics, Hohai University, Nanjing, 210098 China, e-mail: zhouwx2000@163.com
Long Miao, School of Mathematics, Hohai University, Nanjing, 210098 China, e-mail: 20210088@hhu.edu.cn
Baijun Gao, corresponding author, School of Mathematics and Statistics, Yili Normal University, Yining, 835000 China, e-mail: dqgbj2008@163.com
Ran Li, School of Mathematics, Hohai University, Nanjing, 210098 China, e-mail: LIRAN_0901@outlook.com
Cite this article as: Wenxia Zhou, Long Miao, Baijun Gao, Ran Li. A note on the non-solvable formation $\hat{\mathfrak{J}_{pr}}r$. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 4, pp. 290–299.
Русский
Вэнься Чжоу, Лун Мяо, Байцзюнь Гао, Ран Ли. Замечание о неразрешимой формации $\hat{\mathfrak{J}_{pr}}r$
В этой статье мы расширяем формацию $\hat{\mathfrak{J}_{pr}}$, которая порождается классом $\mathfrak{J}_{pr}$, первоначально введенным Деминой и Масловой. Класс $\mathfrak{J}_{pr}$ состоит из конечных групп, в которых каждая неразрешимая максимальная подгруппа имеет примарный индекс. Опираясь на эту структуру, мы вводим и изучаем две обобщенные формации, обозначаемые $\hat{\mathfrak{J}}$ и $\hat{\mathfrak{J}_{p}}$, которые получаются путем включения минимальных неразрешимых максимальных подгрупп и применения локализационного подхода к максимальным подгруппам. Мы устанавливаем новые достаточные условия, при которых конечная группа принадлежит этим формациям. Кроме того, мы приводим примеры неразрешимых групп, иллюстрирующие различия между классом $\mathfrak{J}_{pr}$ и его обобщениями.
Ключевые слова: формация, неразрешимая группа, вторая максимальная подгруппа, ядро подгруппы
