Writing Finite Simple Groups of Lie Type as Products of Subset Conjugates

Abstract

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 \(A\subseteq G\) with \(|A|\ge 2\), 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 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.