发表机构
Alfréd Rényi Institute of Mathematics; King’s College London; Heilbronn Institute for Mathematical Research; University of Warwick（阿尔弗雷德·雷尼数学研究所; 伦敦国王学院; 希尔布隆数学研究所; 华威大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明有限可解群在特定Sylow子群交集条件下存在同步化元素，覆盖奇数阶群并统一旧结果，同时完成对称群、交错群及单群猜想的证明。
AI 中文摘要
设$G$为有限可解群，使得对于任意素数$p$及$G$的任意商群$Q$，$Q$中存在两个Sylow $p$-子群，其交集为$O_p(Q)$。则对于$G$中对应于不同素数$p_1,\ldots,p_n$的任意Sylow子群族$(P_i)_{i=1}^n$，存在$x \in G$使得对所有$i$均有$P_i \cap P_i^x = O_{p_i}(G)$。此结论覆盖奇数阶群，部分解决了第二作者和第四作者提出的猜想，并统一了Bialostocki和Mann关于幂零子群交集的早期结果。我们还证明了适用于所有对称群和交错群的一个结果，完成了由Burness和第一作者发起的对单群猜想的证明。
英文摘要
Let $G$ be a finite solvable group such that, for any prime $p$ and any quotient $Q$ of $G$, there are two Sylow $p$-subgroups of $Q$ intersecting in $O_p(Q)$. Then, for every family $(P_i)_{i=1}^n$ of Sylow subgroups of $G$ for distinct primes $p_1,\ldots,p_n$, there exists $x \in G$ such that $P_i \cap P_i^x = O_{p_i}(G)$ for all $i$. This covers groups of odd order, partially settling a conjecture of the second and fourth authors and unifying old results of Bialostocki and Mann on the intersection of nilpotent subgroups. We also prove a result for all symmetric and alternating groups, completing the proof of the conjecture for simple groups as initiated by Burness and the first author.
Comments11 pages
