On products of conjugacy classes in general linear groups

TL;DR

This paper establishes a precise upper bound for the minimal number of conjugacy class products needed to generate all nontrivial elementary transvections in certain linear groups, with exact values in specific cases.

Contribution

It provides a sharp upper bound for the minimal number of conjugacy class products to generate elementary transvections in linear groups, and determines this number in key cases.

Findings

Established a sharp upper bound for m(C).

Determined m(C) explicitly for algebraically closed fields or n=3 or n=∞.

Extended understanding of conjugacy class products in linear groups.

Abstract

Let $K$ be a field and $n\geq 3$. Let $E_n(K)\leq H\leq GL_n(K)$ be an intermediate group and $C$ a noncentral $H$-class. Define $m(C)$ as the minimal positive integer $m$ such that $\exists i_1,\dots,i_m\in\{\pm 1\}$ such that the product $C^{i_1}\dots C^{i_m}$ contains all nontrivial elementary transvections. In this article we obtain a sharp upper bound for $m(C)$. Moreover, we determine $m(C)$ for any noncentral $H$-class $C$ under the assumption that $K$ is algebraically closed or $n=3$ or $n=\infty$.