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

Daniele Dona
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Abstract

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 Szab\'o [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.
将列类型的有限简单群写成子集共轭的乘积
Liebeck-Nikolov-Shalev猜想[LNS12]断言,对于任意有限简单非阿贝尔群 $G$ 和任意集合 $A\subseteq G$ 且 $|A|\geq 2$,对于某个绝对常数 $N$,$G$ 是 $A$ 的最多 $N\frac\{log|G|}\{log|A|}$ 共轭的乘积。对于 Lie 类型的 $G$,我们证明对于任意 $\varepsilon>0$ 都存在某个 $N_{\varepsilon}$,对于这个 $G$,它是最多 $N_{{varepsilon}\left(\frac\{log|G|}{log|A|}\right)^{1+\varepsilon}$ $A$ 或 $A^{-1}$ 共轭的乘积。对于对称集,这改进了 Liebeck、Nikolov 和 Shalev [LNS12] 以及 Gill、Pyber、Short 和 Szab\'o [GPSS13] 的结果。在本文准备期间,利夫希茨[Lif24]完成了李贝克-尼科洛夫-沙列夫猜想的证明。两篇论文都以 [GLPS24] 为起点。Lifshitz 的论证使用了表征理论的重型机械来完成猜想,而本文则通过相当基本的组合论证获得了一个较为温和的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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