The most symmetric smooth cubic surface over a finite field of characteristic $2$

TL;DR

This paper determines the largest automorphism groups of smooth cubic surfaces over finite fields of characteristic 2, showing they are either PSU}_4(\u00F4_2) or the symmetric group of degree 6, depending on the field size.

Contribution

It classifies the automorphism groups of maximal order for smooth cubic surfaces over finite fields of characteristic 2, identifying unique isomorphism classes.

Findings

Automorphism group is PSU}_4(\u00F4_2) when the field size is a power of 4.

Automorphism group is the symmetric group of degree 6 otherwise.

Such cubic surfaces are unique up to isomorphism.

Abstract

In this paper we find the largest automorphism group of a smooth cubic surface over any finite field of characteristic $2.$ We prove that if the order of the field is a power of $4,$ then the automorphism group of maximal order of~a~smooth cubic surface over this field is $\mathrm{PSU}_4(\mathbb{F}_2).$ If the order of the field of characteristic $2$ is not a power of $4,$ then we prove that the automorphism group of maximal order of a smooth cubic surface over this field is the symmetric group of degree $6.$ Moreover, we prove that smooth cubic surfaces with such properties are unique up to isomorphism.