On products of conjugacy classes in general linear groups - Raimund Preusser (International Journal of Group Theory, 2022)
Let $K$ be a field and $n≥ 3$. Let $E_n(K)≤ H≤ GL_n(K)$ be an intermediate group and $C$ a noncentral $H$-class. Define $m(C)$ as the minimal positive integer $m$ such that $\exists i_1,…,i_m\in\{± 1\}$ such that the product $C^{i_1}… C^{i_m}$ 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 $n=∞$.