Products of involutions in symplectic groups over general fields (II)

TL;DR

This paper extends previous results on expressing elements of symplectic groups as products of involutions from infinite fields to all finite fields (except one special case), providing a comprehensive understanding of involution products in these groups.

Contribution

It adapts and generalizes earlier findings to finite fields, including a detailed analysis of a unique exceptional case where the field size is three and dimension is four.

Findings

Every element of symplectic groups over infinite fields is a product of four or five involutions.

The result is extended to all finite fields except when n=4 and the field size is 3.

A detailed study of the exceptional case where n=4 and | ext{F}|=3.

Abstract

Let $s$ be an $n$-dimensional symplectic form over a field $\mathbb{F}$ of characteristic other than $2$, with $n>2$. In a previous article, we have proved that if $\mathbb{F}$ is infinite then every element of the symplectic group $\mathrm{Sp}(s)$ is the product of four involutions if $n$ is a multiple of $4$ and of five involutions otherwise. Here, we adapt this result to all finite fields with characteristic not $2$, with the sole exception of the very special situation where $n=4$ and $|\mathbb{F}|=3$, a special case which we study extensively.