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

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.