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Writing Finite Simple Groups of Lie Type as Products of Subset Conjugates
被引：0
作者：
Daniele Dona
[1
]
机构：
[1] HUN-REN Alfréd Rényi Institute of Mathematics,
关键词：
Growth;
Conjugacy;
Finite simple groups;
20D06;
20F69;
20G40;
D O I：
10.1007/s00493-025-00155-1
中图分类号：
学科分类号：
摘要：
The Liebeck–Nikolov–Shalev conjecture (Bull Lond Math Soc 44(3):469–472, 2012) asserts that, for any finite simple non-abelian group G and any set A⊆G\documentclass[12pt]{minimal}
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\begin{document}$$A\subseteq G$$\end{document} with |A|≥2\documentclass[12pt]{minimal}
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\begin{document}$$|A|\ge 2$$\end{document}, G is the product of at most Nlog|G|log|A|\documentclass[12pt]{minimal}
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\begin{document}$$N\frac{\log |G|}{\log |A|}$$\end{document} conjugates of A, for some absolute constant N. For G of Lie type, we prove that for any ε>0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon >0$$\end{document} there is some Nε\documentclass[12pt]{minimal}
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\begin{document}$$N_{\varepsilon }$$\end{document} for which G is the product of at most Nεlog|G|log|A|1+ε\documentclass[12pt]{minimal}
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\begin{document}$$N_{\varepsilon }\left( \frac{\log |G|}{\log |A|}\right) ^{1+\varepsilon }$$\end{document} conjugates of either A or A-1\documentclass[12pt]{minimal}
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\begin{document}$$A^{-1}$$\end{document}. For symmetric sets, this improves on results of Liebeck et al. (2012) and Gill et al. (Groups Geom Dyn 7(4):867–882, 2013). During the preparation of this paper, the proof of the Liebeck–Nikolov–Shalev conjecture was completed by Lifshitz (Completing the proof of the Liebeck–Nikolov–Shalev conjecture, 2024, https://arxiv.org/abs/2408.10127). Both papers use Gill et al. (Initiating the proof of the Liebeck–Nikolov–Shalev conjecture, 2024, https://arxiv.org/abs/2408.07800) 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.
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