Products of conjugacy classes in SL₂(k) and PSL₂(k)

Abstract

Let k be a field with u-invariant $\le 2$. Assume further that k is not quadratically closed, $\mathsf{char}(k)\ne 2$ and $\left|k\right|\ge 5$. It is known that the covering number of both ${\text{SL}}_2(k)$ and ${\text{PSL}}_2(k)$ is three, while their extended covering number is four. In this article, we completely describe the product of two conjugacy classes in ${\text{SL}}_2(k)$ and ${\text{PSL}}_2(k)$. Further, we also describe the product of three conjugacy classes (at least two of which are distinct) in ${\text{SL}}_2(k)$ and ${\text{PSL}}_2(k)$.

Keywords:

• Products of conjugacy classes
• projective special linear groups
• special linear groups

2020 Mathematics Subject Classification:

• 20D06
• 20E45

Acknowledgments

We thank Prof. Maneesh Thakur for the argument given in Remark 5.3. We also thank Prof. Amit Kulshrestha for his interest in this work. Finally, we thank the anonymous reviewers for their valuable comments on the article.

Additional information

Funding

The first named author acknowledges the support through Prime Minister’s Research Fellowship from the Ministry of Education, Government of India (PMRF ID: 0601097). The second and third named authors would like to acknowledge the support of IISER Mohali institute post-doctoral fellowship during this work.