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

2024/09/17 by Daniele Dona, Dona, Daniele
Mathematics · #20D06 #20F69 #20G40 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2409.11246

openalex publication_date 2024/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Liebeck-Nikolov-Shalev conjecture [LNS12] asserts that, for any finite simple non-abelian group G and any set A⊆ G with |A|≥ 2, G is the product of at most N(log|G|)/(log|A|) conjugates of A, for some absolute constant N. For G of Lie type, we prove that for any ε>0 there is some N_ε for which G is the product of at most N_ε((log|G|)/(log|A|))^1+ε 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.

Related

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• Completing the proof of the Liebeck--Nikolov--Shalev conjecture
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