发表机构
School of Mathematical Sciences, Peking University（北京大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过抛物子群下降方法与有限单群分解分类，证明了Galt-Tyutyanov猜想，给出有限群为单群的新判据，部分回答Skiba问题，此前仅对部分单群验证过该猜想。
AI 中文摘要
有限群G的子群A是遗传G-置换的，当且仅当对每个满足A≤E≤G的子群E，以及E的每个子群B，B的某个G-共轭在E中与A置换。本文给出有限群为单群的一个新判据：非平凡有限群G是单群当且仅当它没有非平凡真遗传G-置换子群。这证明了Galt和Tyutyanov的猜想，并部分回答了《Kourovka笔记》中问题17.112的(b)部分Skiba提出的问题。此前的工作已对交错群、散在群、李型例外群以及PSL₂(q)和PSU₃(q)群证明了该猜想，本文则对所有典型群证明了该猜想，证明过程利用了通过抛物子群的下降方法以及有限单群的分解分类。
英文摘要
A subgroup $A$ of a finite group $G$ is \emph{hereditarily $G$-permutable} if, for every subgroup $E$ with $A\leq E\leq G$ and every subgroup $B$ of $E$, some $G$-conjugate of $B$ permutes with $A$ in $E$. We give a new criterion for a finite group to be simple. A nontrivial finite group $G$ is simple if and only if it has no proper nontrivial hereditarily $G$-permutable subgroup. This proves the conjecture of Galt and Tyutyanov and answers Skiba's question in part \textup{(b)} of Problem 17.112 in the Kourovka Notebook. Earlier work settled the conjecture for alternating groups, sporadic groups, exceptional groups of Lie type, and the groups ${\rm PSL}_2(q)$ and ${\rm PSU}_3(q)$. We prove the conjecture for all classical groups. The proof uses descent through parabolic subgroups and the classification of factorizations of finite simple groups.
CommentsAI is not used for proving the conjecture. I thank Baoyu Zhang (Birmingham) for suggesting the problem. I am happy to defend every piece of details in the paper via online meeting if anyone interested in this paper contacts me
