On products of conjugacy classes in general linear groups

Document Type : Research Paper


Chebyshev Laboratory, St. Petersburg State University, Russia


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 $\exists i_1,\ldots,i_m\in\{\pm 1\}$ such that the product $C^{i_1}\cdots C^{i_m}$ 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 $n=\infty$.


Main Subjects

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Volume 11, Issue 4 - Serial Number 4
December 2022
Pages 229-252
  • Receive Date: 11 June 2020
  • Revise Date: 13 October 2021
  • Accept Date: 16 October 2021
  • Published Online: 01 December 2022