Abelian Hopf Galois structures on prime-power Galois field extensions
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- by S. C. Featherstonhaugh, A. Caranti and L. N. Childs PDF
- Trans. Amer. Math. Soc. 364 (2012), 3675-3684 Request permission
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 $L|K$ be a Galois extension of fields with abelian Galois group $G$. If also $L|K$ is $H$-Hopf Galois, where the $K$-Hopf algebra $H$ has associated group $N$ with $N$ as above, then $N$ is isomorphic to $G$.References
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Additional Information
- 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
- MR Author ID: 45160
- ORCID: 0000-0002-5746-9294
- Email: caranti@science.unitn.it
- L. N. Childs
- Affiliation: Department of Mathematics, University at Albany, Albany, New York 12222
- Email: lchilds@albany.edu
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2901229