Non-existence of Hopf-Galois structures and bijective crossed homomorphisms

TL;DR

This paper investigates the non-existence of certain Hopf-Galois structures on Galois extensions by analyzing regular subgroups of holomorphs and their relation to bijective crossed homomorphisms.

Contribution

It provides a new description of regular subgroups of holomorphs in terms of bijective crossed homomorphisms and applies this to study non-existence results for Hopf-Galois structures.

Findings

Characterization of regular subgroups via bijective crossed homomorphisms

Conditions under which Hopf-Galois structures do not exist

Insights into the structure of holomorphs and their subgroups

Abstract

By work of C. Greither and B. Pareigis as well as N. P. Byott, the enumeration of Hopf-Galois structures on a Galois extension of fields with Galois group $G$ may be reduced to that of regular subgroups of $\mbox{Hol}(N)$ isomorphic to $G$ as $N$ ranges over all groups of order $|G|$, where $\mbox{Hol}(-)$ denotes the holomorph. In this paper, we shall give a description of such subgroups of $\mbox{Hol}(N)$ in terms of bijective crossed homomorphisms $G\longrightarrow N$, and then use it to study two questions related to non-existence of Hopf-Galois structures.