Hopf-Galois structures on finite extensions with almost simple Galois group

TL;DR

This paper investigates Hopf-Galois structures on finite Galois extensions with almost simple Galois groups, providing criteria for associated groups and counting structures in specific cases.

Contribution

It introduces necessary group-theoretic criteria for Hopf-Galois structures and determines their number for extensions with socle of prime index.

Findings

Criteria for groups N associated with Hopf-Galois structures

Number of structures for Galois groups with socle A×C_p

Characterization of structures in almost simple groups

Abstract

In this paper, we study the Hopf-Galois structures on a finite Galois extension whose Galois group $G$ is an almost simple group in which the socle $A$ has prime index $p$. Each Hopf-Galois structure is associated to a group $N$ of the same order as $G$. We give necessary criteria on these $N$ in terms of their group-theoretic properties, and determine the number of Hopf-Galois structures associated to $A\times C_p$, where $C_p$ is the cyclic group of order $p$.