On the product decomposition conjecture for finite simple groups

TL;DR

This paper proves a bound on how many conjugates of a subset can generate a finite simple group of Lie type, confirming a conjecture for groups of bounded rank and providing insights into their product structure.

Contribution

It establishes a logarithmic bound on the number of conjugates needed to express such groups, confirming a key conjecture for bounded rank Lie type groups.

Findings

Finite simple groups of Lie type can be expressed as a product of a logarithmic number of conjugates of a subset.

The bound depends only on the Lie rank of the group.

The result confirms a conjecture by Liebeck, Nikolov, and Shalev for bounded rank cases.

Abstract

We prove that if $G$ is a finite simple group of Lie type and $S$ a subset of $G$ of size at least two then $G$ is a product of at most $c\log|G|/\log|S|$ conjugates of $S$, where $c$ depends only on the Lie rank of $G$. This confirms a conjecture of Liebeck, Nikolov and Shalev in the case of families of simple groups of Lie type of bounded rank.