Hopf Galois structures on symmetric and alternating extensions

TL;DR

This paper characterizes the types of Hopf Galois structures present on Galois extensions with symmetric or alternating Galois groups, building on previous results about induced structures and normality conditions.

Contribution

It provides a classification of Hopf Galois structures for extensions with symmetric and alternating Galois groups, extending prior theoretical frameworks.

Findings

Identifies which Hopf Galois structures occur for symmetric group Galois extensions.

Determines the types of structures for alternating group Galois extensions.

Uses recent results on normality and induced structures to achieve classification.

Abstract

By using our previous results on induced Hopf Galois structures and a recent result by Koch, Kohl, Truman and Underwood on normality, we determine which types of Hopf Galois structures occur on Galois extensions with Galois group isomorphic to alternating or symmetric groups.