The Multiplicative Automorphisms of a Finite Nearfield, with an Application

TL;DR

This paper investigates automorphisms of finite nearfields' multiplicative groups, develops counting methods for nearfields, and applies these to enumerate near vector spaces up to isomorphism.

Contribution

It provides new partial results on automorphism groups of finite nearfields and introduces counting techniques for nearfields and near vector spaces.

Findings

Partial automorphism group characterizations

Counting formulas for finite nearfields

Enumeration of near vector spaces up to isomorphism

Abstract

In this paper we look at the automorphisms of the multiplicative group of finite nearfields. We find partial results for the actual automorphism groups. We find counting techniques for the size of all finite nearfields. We then show that these results can be used in order to count the number of near vector spaces of a given dimension over a given nearfield, up to isomorphism.