Skew Braces and Hopf-Galois Structures of Heisenberg Type

TL;DR

This paper classifies skew braces of Heisenberg type for primes greater than 3, determines their automorphism groups, and uses this to identify all related Hopf-Galois structures on degree p^3 Galois extensions.

Contribution

It provides a complete classification of skew braces of Heisenberg type and links this to Hopf-Galois structures, advancing understanding of algebraic structures in field extensions.

Findings

Classified all skew braces of Heisenberg type for p>3

Determined automorphism groups of these skew braces

Identified all Hopf-Galois structures of Heisenberg type on degree p^3 extensions

Abstract

We classify all skew braces of Heisenberg type for a prime number $ p>3 $. Furthermore, we determine the automorphism group of each one of these skew braces (as well as their socle and annihilator). Hence, by utilising a link between skew braces and Hopf-Galois theory, we can determine all Hopf-Galois structures of Heisenberg type on Galois field extensions of fields of degree $ p^{3} $.