Characteristic Subgroup Lattices and Hopf-Galois Structures

TL;DR

This paper explores the structure of Hopf-Galois extensions, focusing on the relationship between regular subgroups, characteristic subgroups, and Galois correspondence, revealing conditions under which certain structures cannot exist.

Contribution

It introduces a new correspondence between characteristic subgroups of regular subgroups and subgroups of the Galois group, and uses it to identify when certain Hopf-Galois structures are impossible.

Findings

Characterizes the relationship between characteristic subgroups and Galois subgroups.

Shows that some collections of regular subgroups must be empty for certain group pairings.

Provides criteria for the non-existence of specific Hopf-Galois structures.

Abstract

The Hopf-Galois structures on normal extensions $K/k$ with $G=Gal(K/k)$ are in one-to-one correspondence with the set of regular subgroups $N\leq B=Perm(G)$ that are normalized by the left regular representation $\lambda(G)\leq B$. Each such $N$ corresponds to a Hopf algebra $H_N=(K[N])^G$ that acts on $K/k$. Such regular subgroups $N$ need not be isomorphic to $G$ but must have the same order. One can subdivide the totality of all such $N$ into collections $R(G,[M])$ which is the set of those regular $N$ normalized by $\lambda(G)$ and isomorphic to a given abstract group $M$ where $|M|=|G|$. There arises an injective correspondence between the characteristic subgroups of a given $N$ an d the set of subgroups of $G$ stemming from the Galois correspondence between sub-Hopf algebras of $H_N$ and intermediate fields $k\subseteq F\subseteq K$. We utilize this correspondence to show that for certain pairings $(G,[M])$, the collection $R(G,[M])$ must be empty.