Abelian Hopf Galois structures on prime-power Galois field extensions. (English) Zbl 1287.12002

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.


12F10 Separable extensions, Galois theory
16T05 Hopf algebras and their applications
Full Text: DOI


[1] N. P. Byott, Uniqueness of Hopf Galois structure for separable field extensions, Comm. Algebra 24 (1996), no. 10, 3217 – 3228. , https://doi.org/10.1080/00927879608825743 N. P. Byott, Corrigendum: ”Uniqueness of Hopf Galois structure for separable field extensions”, Comm. Algebra 24 (1996), no. 11, 3705. · Zbl 0878.12001 · doi:10.1080/00927879608825779
[2] Nigel P. Byott, Hopf-Galois structures on field extensions with simple Galois groups, Bull. London Math. Soc. 36 (2004), no. 1, 23 – 29. · Zbl 1038.12002 · doi:10.1112/S0024609303002595
[3] A. Caranti, F. Dalla Volta, and M. Sala, Abelian regular subgroups of the affine group and radical rings, Publ. Math. Debrecen 69 (2006), no. 3, 297 – 308. · Zbl 1123.20002
[4] Scott Carnahan and Lindsay Childs, Counting Hopf Galois structures on non-abelian Galois field extensions, J. Algebra 218 (1999), no. 1, 81 – 92. · Zbl 0988.12003 · doi:10.1006/jabr.1999.7861
[5] Stephen U. Chase and Moss E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in Mathematics, Vol. 97, Springer-Verlag, Berlin-New York, 1969. · Zbl 0197.01403
[6] Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, Mathematical Surveys and Monographs, vol. 80, American Mathematical Society, Providence, RI, 2000. · Zbl 0944.11038
[7] Lindsay N. Childs, On Hopf Galois structures and complete groups, New York J. Math. 9 (2003), 99 – 115. · Zbl 1038.12003
[8] Lindsay N. Childs, Elementary abelian Hopf Galois structures and polynomial formal groups, J. Algebra 283 (2005), no. 1, 292 – 316. · Zbl 1071.16031 · doi:10.1016/j.jalgebra.2004.07.009
[9] Lindsay N. Childs, Some Hopf Galois structures arising from elementary abelian \?-groups, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3453 – 3460. · Zbl 1128.16022
[10] S. C. Featherstonhaugh, Abelian Hopf Galois extensions on Galois field extensions of prime power order, Ph.D. thesis, Univ. at Albany, NY, 2003. · Zbl 1287.12002
[11] Cornelius Greither and Bodo Pareigis, Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), no. 1, 239 – 258. · Zbl 0615.12026 · doi:10.1016/0021-8693(87)90029-9
[12] Christopher J. Hillar and Darren L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no. 10, 917 – 923. · Zbl 1156.20046
[13] N. Jacobson, Representation theory for Jordan rings, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R. I., 1952, pp. 37 – 43.
[14] Timothy Kohl, Classification of the Hopf Galois structures on prime power radical extensions, J. Algebra 207 (1998), no. 2, 525 – 546. · Zbl 0953.12003 · doi:10.1006/jabr.1998.7479
[15] Timothy Kohl, Groups of order 4\?, twisted wreath products and Hopf-Galois theory, J. Algebra 314 (2007), no. 1, 42 – 74. · Zbl 1129.16031 · doi:10.1016/j.jalgebra.2007.04.001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.