循环2-子群正规化子的可解补充
Solvable Supplements to Normalizers of Cyclic 2-Subgroups
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中文总结 AI 辅助
本文放宽了可解补充条件至仅对循环2-子群成立，刻画了满足该条件的有限群的非交换合成因子为特定PSL2(q)族，并给出可解性的充要条件。
中文摘要 AI 辅助
Amberg和Kazarin证明了：若有限群的每个素数幂阶循环子群的正规化子都有一个可解补充，则该群是可解的。我们通过仅要求循环$2$-子群满足此条件，大幅放宽了这一假设。此条件记为$\mathrm{SSN}_2$，它将群的非交换合成因子严格限制为$\PSL_2(q)$族，其中$q\geq7$是满足$q\equiv3\pmod4$的素数幂。反之，该族恰好是满足$\mathrm{SSN}_2$的非交换有限单群。因此，满足$\mathrm{SSN}_2$的有限群是可解的，当且仅当它没有同构于这些群之一的截面。
英文摘要
Amberg and Kazarin proved that a finite group is solvable if the normalizer of every cyclic subgroup of prime power order has a solvable supplement. We substantially relax this hypothesis by requiring it only for cyclic $2$-subgroups. This condition, denoted by $\mathrm{SSN}_2$, sharply restricts the nonabelian composition factors of the group to the family $\PSL_2(q)$, where $q\geq7$ is a prime power satisfying $q\equiv3\pmod4$. Conversely, this family is precisely the nonabelian finite simple groups that satisfy $\mathrm{SSN}_2$. Consequently, a finite group satisfying $\mathrm{SSN}_2$ is solvable if and only if it has no section isomorphic to one of these groups.
发表机构
- School of Mathematics and Statistics Guangdong University of Technology（广东工业大学数学与统计学院）
- School of Mathematics and Statistics The University of Melbourne（墨尔本大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
