Initiating the proof of the Liebeck--Nikolov--Shalev conjecture

TL;DR

This paper proves a significant case of the Liebeck--Nikolov--Shalev conjecture, showing that for sufficiently large subsets of finite simple groups, a bounded number of conjugates can generate the entire group.

Contribution

It establishes the conjecture for subsets larger than a fixed power of the group size and proves a Skew Product Theorem for all finite simple groups.

Findings

The conjecture holds when |A| > |S|^c for some constant c.

Either the product of two conjugates of A grows substantially or a bounded number of conjugates cover S.

The Skew Product Theorem is valid for all finite simple groups.

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.