arXiv 2409.112462024-09-17math.GRmath.CO

将有限单李群表示为子集共轭的乘积

Writing finite simple groups of Lie type as products of subset conjugates

Daniele Dona

备注：: 13 pages

阅读 PDF

arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文针对有限单李群，证明了对任意ε>0，存在N_ε，使群可表为至多N_ε(log|G|/log|A|)^{1+ε}个A或A^{-1}的共轭的乘积，改进了对称集下的已有结果，采用初等组合论证，与同期用表示论证明LNS猜想的工作形成对照。

摘要：

Liebeck-Nikolov-Shalev猜想[LNS12]断言，对任意有限非阿贝尔单群$G$和任意满足$|A|\geq2$的集合$A\subseteq G$，存在某个绝对常数$N$，使得$G$可表示为至多$N\frac{\log|G|}{\log|A|}$个$A$的共轭的乘积。对于李型的$G$，我们证明对任意$\varepsilon>0$，存在某个$N_{\varepsilon}$，使得$G$可表示为至多$N_{\varepsilon}\left(\frac{\log|G|}{\log|A|}\right)^{1+\varepsilon}$个$A$或$A^{-1}$的共轭的乘积。对于对称集，这一结果改进了Liebeck、Nikolov、Shalev[LNS12]以及Gill、Pyber、Short、Szabó[GPSS13]的结论。在本文撰写期间，Lifshitz[Lif24]完成了Liebeck-Nikolov-Shalev猜想的证明。两篇论文均以[GLPS24]为起点。Lifshitz的论证运用了表示论中的高深工具来完成该猜想，而本文通过相当初等的组合论证得到了一个相对温和的结果。

英文摘要：

The Liebeck-Nikolov-Shalev conjecture [LNS12] asserts that, for any finite simple non-abelian group $G$ and any set $A\subseteq G$ with $|A|\geq 2$, $G$ is the product of at most $N\frac{\log|G|}{\log|A|}$ conjugates of $A$, for some absolute constant $N$. For $G$ of Lie type, we prove that for any $\varepsilon>0$ there is some $N_{\varepsilon}$ for which $G$ is the product of at most $N_{\varepsilon}\left(\frac{\log|G|}{\log|A|}\right)^{1+\varepsilon}$ conjugates of either $A$ or $A^{-1}$. For symmetric sets, this improves on results of Liebeck, Nikolov, and Shalev [LNS12] and Gill, Pyber, Short, and Szabó [GPSS13]. During the preparation of this paper, the proof of the Liebeck-Nikolov-Shalev conjecture was completed by Lifshitz [Lif24]. Both papers use [GLPS24] as a starting point. Lifshitz's argument uses heavy machinery from representation theory to complete the conjecture, whereas this paper achieves a more modest result by rather elementary combinatorial arguments.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

AI总结：

摘要：

英文摘要：

相关论文