Random unipotent Sylow subgroups of groups of Lie type of bounded rank

TL;DR

This paper demonstrates that finite simple groups of Lie type with bounded rank can be expressed as a product of 11 random unipotent Sylow subgroups with high probability, without relying on the classification theorem.

Contribution

It establishes a new probabilistic product decomposition result for finite simple groups of Lie type with bounded rank, reducing the number of Sylow subgroups needed.

Findings

Finite simple groups of Lie type are products of 11 random unipotent Sylow subgroups with high probability.

The result holds as the group size tends to infinity.

The proofs do not depend on the classification of finite simple groups.

Abstract

In 2001 Liebeck and Pyber showed that a finite simple group of Lie type is a product of $ 25 $ carefully chosen unipotent Sylow subgroups. Later, in a series of works it was shown that $ 4 $ unipotent Sylow subgroups suffice. We prove that if the rank of a finite simple group of Lie type $ G $ is bounded, then $ G $ is a product of $ 11 $ random unipotent Sylow subgroups with probability tending to $ 1 $ as $ |G| $ tends to infinity. An application of the result to finite linear groups is given. The proofs do not depend on the classification of finite simple groups.