Counting Hopf Galois structures on non-abelian Galois field extensions
If \(L/K\) is a \(G\)-Galois extension of fields, then it is quite possible that \(L/K\) is also Hopf Galois for various Hopf algebras \(H\) which are different from the obvious choice \(H=K[G]\). In particular it was observed by Pareigis and the reviewer that such a ``genuinely Hopf'' Galois structure exists as soon as \(G\) is not abelian. The paper under review provides information on the number \(h(G)\) of such Hopf structures for some particular nonabelian groups \(G\). (The number \(h(G)\) is indeed independent of \(L/K\), since the whole theory of separable Hopf Galois extensions can be formulated exclusively in terms of groups and actions.) One corollary is that for \(G=S_n\) and \(n\geq 5\) there are at least \(\sqrt{n!}\) of them. A little more precisely: The number \(n(G)\) is the sum of the numbers \(e(G,N)\) (whose definition we omit) over all groups \(N\) of same order as \(G\). For given \(G\), the question for which \(N\) the number \(e(G,N)\) is nonzero seems to be a tricky one. In the situation at hand, \(e(S_n,S_n)\) and \(e(S_n,A_n\times C_2)\) are both calculated exactly, which gives the above lower bound for \(h(S_n)\). The methods are essentially elementary, but they do invoke quite a few nontrivial results from group theory.
scientific article; zbMATH DE number 1340572