Products of involutions in symplectic groups I: bireflections

TL;DR

This paper classifies elements that are products of two involutions and are reversible in symplectic groups over various fields, providing a detailed understanding of their structure and properties.

Contribution

It offers a comprehensive classification of bireflectional and reversible elements in symplectic groups and their projective counterparts, extending previous work in the field.

Findings

Classification of bireflectional elements in symplectic groups

Identification of reversible elements in finite projective symplectic groups

Structural insights into involutions in symplectic groups

Abstract

We classify bireflectional elements (products of 2 involutions) in symplectic groups Sp$(2n, K)$ over a field $K$. We also classify rev ersible elements (elements conjugate to their inverses) and bireflectional elements in finite projective symplectic groups PSp$(2n,q)$.