On products of conjugacy classes in general linear groups

Raimund Preusser

paper

On products of conjugacy classes in general linear groups

arXiv:2006.06502

Abstract

Let $K$ be a field and $n \geq 3$ . Let $E_{n} (K) \leq H \leq G L_{n} (K)$ 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 $n = \infty$ .