Groups equal to a product of three conjugate subgroups

TL;DR

This paper proves that any finite non-solvable group can be expressed as a product of three conjugates of a proper subgroup, improving previous bounds and introducing new methods involving double cosets and BN-pairs.

Contribution

It establishes the minimal number of conjugate subgroup products needed to cover non-solvable groups and introduces a novel proof technique using dioids and double coset analysis.

Findings

Any finite non-solvable group is the product of three conjugates of a proper subgroup.

A new bound reducing the number of conjugates from 36 to 3.

A general theorem for groups with BN-pairs and finite Weyl groups.

Abstract

Let $G$ be a finite non-solvable group. We prove that there exists a proper subgroup $A$ of $G$ such that $G$ is the product of three conjugates of $A$, thus replacing an earlier upper bound of $36$ with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group $G$ with a $BN$-pair and a finite Weyl group $W$ satisfies $G=\left( Bn_{0}B\right) ^{2}=BB^{n_{0}}B$ where $n_{0}$ is any preimage of the longest element of $W$. The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of $B$ in $G$. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.