Product decompositions in finite simple groups

TL;DR

This paper investigates how finite simple groups can be expressed as products of conjugates of subsets, proving the conjecture for bounded subsets and large subsets of Lie type groups, and introducing new product decompositions.

Contribution

It proves a general conjecture on product decompositions in finite simple groups for specific subset sizes and introduces new decomposition methods leveraging recent growth theory advances.

Findings

Proved the conjecture for bounded subsets in all finite simple groups.

Established the conjecture for large subsets in Lie type groups of bounded rank.

Developed new product decompositions using recent growth theory results.

Abstract

We propose a general conjecture on decompositions of finite simple groups as products of conjugates of an arbitrary subset. We prove this conjecture for bounded subsets of arbitrary finite simple groups, and for large subsets of groups of Lie type of bounded rank. Some of our arguments apply recent advances in the theory of growth in finite simple groups of Lie type, and provide a variety of new product decompositions of these groups.