Initiating the proof of the Liebeck--Nikolov--Shalev conjecture
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913516650233856 |
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| author | Gill, Nick Lifshitz, Noam Pyber, László Szabó, Endre |
| author_facet | Gill, Nick Lifshitz, Noam Pyber, László Szabó, Endre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07800 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Initiating the proof of the Liebeck--Nikolov--Shalev conjecture Gill, Nick Lifshitz, Noam Pyber, László Szabó, Endre Group Theory 20D06 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. |
| title | Initiating the proof of the Liebeck--Nikolov--Shalev conjecture |
| topic | Group Theory 20D06 |
| url | https://arxiv.org/abs/2408.07800 |