Abelian Hopf Galois structures on prime-power Galois field extensions (Q2839950)

Let \(L/K\) be a Galois field extension with finite Galois group \(G\) and a Hopf Galois extension with a \(K\)-Hopf algebra \(H\). Then \(L\otimes_KH \cong LN\), a group ring of a regular subgroup \(N\) of \(\text{perm}(G)\) normalized by \(\lambda(G)\), where \(\lambda (G)\) is the image of the left regular representation of \(G\) in \(\text{perm}(G)\). The group \(N\) is then called the associated group of \(H\). Let \(e(G,N)\) be the number of equivalent classes of regular embeddings of \(G\) into the holomorph \(\text{Hol}(N)\) of \(N\), where two embeddings \(\beta\), \(\beta^\prime: G \rightarrow \text{Hol}(N)\) are equivalent if there is an automorphism \(\gamma\) of \(N\) such that for all \(\sigma \in G\), \(\gamma \beta (\sigma) \gamma^{-1} = \beta^\prime(\sigma)\). Then the number of Hopf Galois structures on \(L/K\) is the sum \(\sum e(G,N)\) where the sum is over all isomorphism types of groups \(N\) of the same order as \(G\). The authors show that if \(G\) and \(N\) are non-isomorphic abelian \(p\)-groups for a prime number \(p\) such that \(N\) has \(p\)-rank \(m\) and \(p > m+1\), then \(e(G, N) = 0\). This is a consequence of the following theorem: Let \(p\) be prime and \(N\) a finite abelian \(p\)-group of \(p\)-rank \(m\). If \(m+1 < p,\) then every regular abelian subgroup of \(\text{Hol}(N)\) is isomorphic to \(N\). Several examples are given to show that the hypotheses in the above theorem are necessary.