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

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 $G⟶N$, and then use it to study two questions related to non-existence of Hopf-Galois structures.

Accepted version

References in corpus (1)

• Solubility Criteria for Hopf-Galois Structures

Cited by in corpus (12)

• On the solvability of regular subgroups in the holomorph of a finite solvable group
• On the multiple holomorph of a finite almost simple group
• Hopf-Galois structures on finite extensions with almost simple Galois group
• Hopf-Galois structures of isomorphic type on a non-abelian characteristically simple extension
• Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group
• Hopf-Galois structures on finite extensions with quasisimple Galois group
• Hopf-Galois structures on a Galois $S_n$-extension
• On the ranks of the additive and the multiplicative groups of a brace
• The multiple holomorph of centerless groups
• Non-abelian simple groups which occur as the type of a Hopf-Galois structure on a solvable extension
• Finite skew braces with isomorphic non-abelian characteristically simple additive and circle groups
• Extreme Reidemeister spectra of finite groups