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Abelian Hopf Galois structures on prime-power Galois field extensions


Authors: S. C. Featherstonhaugh, A. Caranti and L. N. Childs
Journal: Trans. Amer. Math. Soc. 364 (2012), 3675-3684
MSC (2010): Primary 12F10; Secondary 16N20, 20B25, 16W30
DOI: https://doi.org/10.1090/S0002-9947-2012-05503-6
Published electronically: March 8, 2012
MathSciNet review: 2901229
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Abstract: The main theorem of this paper is that if $ (N, +)$ is a finite abelian $ p$-group of $ p$-rank $ m$ where $ m+1< p$, then every regular abelian subgroup of the holomorph of $ N$ is isomorphic to $ N$. The proof utilizes a connection, observed by Caranti, Dalla Volta, and Sala, between regular abelian subgroups of the holomorph of $ N$ and nilpotent ring structures on $ (N, +)$. Examples are given that limit possible generalizations of the theorem. The primary application of the theorem is to Hopf Galois extensions of fields. Let $ L\vert K$ be a Galois extension of fields with abelian Galois group $ G$. If also $ L\vert K$ is $ H$-Hopf Galois, where the $ K$-Hopf algebra $ H$ has associated group $ N$ with $ N$ as above, then $ N$ is isomorphic to $ G$.


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S. C. Featherstonhaugh
Affiliation: Department of Mathematics, Borough of Manhattan Community College/CUNY, 199 Chambers Street, Room N-520, New York, New York 10007
Email: sfeatherstonhaugh@bmcc.cuny.edu

A. Caranti
Affiliation: Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, I-38123 Trento, Italy
Email: caranti@science.unitn.it

L. N. Childs
Affiliation: Department of Mathematics, University at Albany, Albany, New York 12222
Email: lchilds@albany.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05503-6
Received by editor(s): July 1, 2010
Received by editor(s) in revised form: August 24, 2010, and November 12, 2010
Published electronically: March 8, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.