发表机构
Universitat Jaume I（豪梅一世大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究《库罗夫卡笔记本》中关于极大子群的问题，对于奇数\(p\)，证明满足特定条件的非单非可解群是\(p\)-幂零群和\(p\)-可分解群，\(p = 2\)时改进相关结果，给出特定非可解群的完整分类。
AI 中文摘要
设\(p\)为素数。当\(p\)为奇数时，研究每个极大子群要么是非阿贝尔单群要么是\(p\)-幂零群，以及每个极大子群要么是非阿贝尔单群要么是\(p\)-可分解群的有限群。证明了满足这些条件的非单、非可解群分别是\(p\)-幂零群和\(p\)-可分解群。回答了《库罗夫卡笔记本》中两个开放问题。当\(p = 2\)时，改进了相关主要结果，给出极大子群为非阿贝尔单群或\(2\)-幂零群的非可解群的完整分类。
英文摘要
Let $p$ be a prime number. When $p$ is odd, we study finite groups in which every maximal subgroup is either non-abelian simple or $p$-nilpotent, as well as those in which every maximal subgroup is either non-abelian simple or $p$-decomposable. We prove that every non-simple, non-solvable group satisfying the first condition is $p$-nilpotent, and every non-simple, non-solvable group satisfying the second condition is $p$-decomposable. In addition, we determine the possible occurrence of non-abelian simple maximal subgroups in these cases. These results provide a substantial partial answer to two questions posed by V.S. Monakhov and I.N. Tyutyanov in the Kourovka Notebook concerning the non-abelian composition factors of such groups. For non-abelian simple groups, we determine those satisfying the corresponding conditions within the alternating and sporadic families. The case of simple groups of Lie type is left open. Finally, for $p=2$, we obtain a complete classification of the non-solvable finite groups whose maximal subgroups are either non-abelian simple or $2$-nilpotent.
Comments17 pages
