Automorphism Groups of some Finite Semifields

TL;DR

This paper computes automorphism groups for specific finite semifields, including Sandler's, Hughes-Kleinfeld, and Knuth semifields, and explores their classification over finite fields.

Contribution

It provides explicit automorphism groups for several well-known finite semifields and analyzes their classification based on field extensions.

Findings

Automorphism group of Sandler's semifields determined

Automorphism groups of Hughes-Kleinfeld and Knuth semifields computed in certain cases

Number of nonisomorphic Sandler semifields constructed from finite fields analyzed

Abstract

We determine the automorphism group for some well known constructions of finite semifields. In particular, we compute the automorphism group of Sandler's semifields and in certain cases the automorphism groups of the Hughes-Kleinfeld and Knuth semifields. We also determine how many nonisomorphic Sandler semifields can be constructed given a finite field F and a finite extension L/F.