Counting Hopf Galois structures on non-abelian Galois field extensions
type: Item
instance of: scholarly article
title: Counting Hopf Galois structures on non-abelian Galois field extensions
zbMATH Open document ID: 0988.12003
author: Scott Carnahan
author: Lindsay N. Childs
published in: Journal of Algebra
publication date: 2000-05-04T00:00:00Z
review text: 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.
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 1340572
zbMATH Keywords: Hopf Galois extensions
zbMATH Keywords: holomorphs
zbMATH Keywords: symmetric groups
describes a project that uses: GAP
MaRDI profile type: Publication
cites work: Uniqueness of hopf galois structure for separable field extensions
cites work: Hopf algebras and Galois theory
cites work: Q3941595
cites work: Hopf Galois theory for separable field extensions
cites work: Classification of the Hopf Galois structures on prime power radical extensions
cites work: Combinatorial structure on the automorphism group of \(S_ 6\)
cites work: Q3941589
description: scientific article; zbMATH DE number 1340572