Elation groups of the Hermitian surface $H(3,q^2)$ over a finite field of characteristic 2

Abstract

Let $\mathcal{S}=\left(\mathcal{P},\mathcal{ℬ},\mathcal{ℐ}\right)$ be a finite generalized quadrangle having order ($s,t$). Let $p$ be a point of $\mathcal{S}$. A whorl about $p$ is a collineation of $\mathcal{S}$ fixing all the lines through $p$. An elation about $p$ is a whorl that does not fix any point not collinear with $p$, or is the identity. If $\mathcal{S}$ has an elation group acting regularly on the set of points not collinear with $p$ we say that $\mathcal{S}$ is an elation generalized quadrangle with base point $p$. The following question has been posed: Can there be two non-isomorphic elation groups about the same point $p$? In this presentation, we show that there are exactly two (up to isomorphism) elation groups of the Hermitian surface $H\left(3,q^2\right)$ over a finite field of characteristic 2.