Characteristic Subgroup Lattices and 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.