Completing the proof of the Liebeck--Nikolov--Shalev conjecture
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912046966112256 |
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| author | Lifshitz, Noam |
| author_facet | Lifshitz, Noam |
| contents | Liebeck, Nikolov, and Shalev conjectured the existence of an absolute constant $C>0$, such that for every subset $A$ of a finite simple group $G$ with $|A|\ge 2$, there exists $C\log|G|/\log|A|$ conjugates of $A$ whose product is $G$. This paper is a companion to \cite{GLPS}, and together they prove the conjecture.
To prove the conjecture, we establish the following skew-product theorem. We show that there exists $ c > 0 $ such that for all $ ε> 0 $ and subsets $ A, B \subseteq G $ of finite simple groups of Lie type, if $ |B| < |G|^{1 - ε} $, then $ |A^σ B| > |B||A|^{c ε} $ for some $ σ\in G $. This result, along with its more involved analogue for alternating groups, constitutes the main contribution of this paper.
Our proof leverages deep results from character theory alongside the probabilistic method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Completing the proof of the Liebeck--Nikolov--Shalev conjecture Lifshitz, Noam Group Theory Combinatorics Liebeck, Nikolov, and Shalev conjectured the existence of an absolute constant $C>0$, such that for every subset $A$ of a finite simple group $G$ with $|A|\ge 2$, there exists $C\log|G|/\log|A|$ conjugates of $A$ whose product is $G$. This paper is a companion to \cite{GLPS}, and together they prove the conjecture. To prove the conjecture, we establish the following skew-product theorem. We show that there exists $ c > 0 $ such that for all $ ε> 0 $ and subsets $ A, B \subseteq G $ of finite simple groups of Lie type, if $ |B| < |G|^{1 - ε} $, then $ |A^σ B| > |B||A|^{c ε} $ for some $ σ\in G $. This result, along with its more involved analogue for alternating groups, constitutes the main contribution of this paper. Our proof leverages deep results from character theory alongside the probabilistic method. |
| title | Completing the proof of the Liebeck--Nikolov--Shalev conjecture |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2408.10127 |