On the quasi group of a cubic surface over a finite field

TL;DR

This paper investigates the structure of the quasi group associated with certain cubic surfaces over finite fields, constructing specific homomorphisms and classifying their uniqueness based on Cayley's classification.

Contribution

It constructs and classifies all possible nontrivial homomorphisms from the quasi group of cubic surfaces over finite fields, confirming their uniqueness in certain cases.

Findings

Constructed explicit nontrivial homomorphisms for some cubic surfaces.

Proved that these are the only such homomorphisms in those cases.

Confirmed the absence of nontrivial homomorphisms in other cases.

Abstract

We construct nontrivial homomorphisms from the quasi group of some cubic surfaces over $\bbF_{\!p}$ into a group. We show experimentally that the homomorphisms constructed are the only possible ones and that there are no nontrivial homomorphisms in the other cases. Thereby, we follow the classification of cubic surfaces, due to A. Cayley.