Initiating the proof of the Liebeck--Nikolov--Shalev conjecture
Abstract
Liebeck, Nikolov, and Shalev conjectured that for every subset A of a finite simple group S with |A|>1, there exist O( log|S| / log|A| ) conjugates of A whose product is S. This paper is a companion to [Lifshitz: Completing the proof of the Liebeck-Nikolov-Shalev conjecture] and together they prove the conjecture. In this paper we prove the conjecture in the regime where $|A|>|S|^c$ for an absolute constant c>0. We also prove that the following Skew Product Theorem holds for all finite simple groups. Namely we show that either the product of two conjugates of A has size at least $|A|^{1.49}$, or S is the product of boundedly many conjugates of A.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.07800
- arXiv:
- arXiv:2408.07800
- Bibcode:
- 2024arXiv240807800G
- Keywords:
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- Mathematics - Group Theory;
- 20D06
- E-Print:
- Citation added. Supporting grant numbers corrected