Fixed-point free endomorphisms and Hopf Galois structures
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Abstract:
Let $L|K$ be a Galois extension of fields with finite Galois group $G$. Greither and Pareigis showed that there is a bijection between Hopf Galois structures on $L|K$ and regular subgroups of $Perm(G)$ normalized by $G$, and Byott translated the problem into that of finding equivalence classes of embeddings of $G$ in the holomorph of groups $N$ of the same cardinality as $G$. In 2007 we showed, using Byott’s translation, that fixed point free endomorphisms of $G$ yield Hopf Galois structures on $L|K$. Here we show how abelian fixed point free endomorphisms yield Hopf Galois structures directly, using the Greither-Pareigis approach and, in some cases, also via the Byott translation. The Hopf Galois structures that arise are “twistings” of the Hopf Galois structure by $H_{\lambda }$, the $K$-Hopf algebra that arises from the left regular representation of $G$ in $Perm(G)$. The paper concludes with various old and new examples of abelian fixed point free endomorphisms.References
- N. P. Byott, Uniqueness of Hopf Galois structure for separable field extensions, Comm. Algebra 24 (1996), no. 10, 3217–3228. MR 1402555, DOI 10.1080/00927879608825743
- Nigel P. Byott, Integral Hopf-Galois structures on degree $p^2$ extensions of $p$-adic fields, J. Algebra 248 (2002), no. 1, 334–365. MR 1879021, DOI 10.1006/jabr.2001.9053
- Nigel P. Byott, Hopf-Galois structures on field extensions with simple Galois groups, Bull. London Math. Soc. 36 (2004), no. 1, 23–29. MR 2011974, DOI 10.1112/S0024609303002595
- Scott Carnahan and Lindsay Childs, Counting Hopf Galois structures on non-abelian Galois field extensions, J. Algebra 218 (1999), no. 1, 81–92. MR 1704676, DOI 10.1006/jabr.1999.7861
- 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. MR 1767499, DOI 10.1090/surv/080
- Lindsay N. Childs, On Hopf Galois structures and complete groups, New York J. Math. 9 (2003), 99–115. MR 2016184
- Lindsay N. Childs, Elementary abelian Hopf Galois structures and polynomial formal groups, J. Algebra 283 (2005), no. 1, 292–316. MR 2102084, DOI 10.1016/j.jalgebra.2004.07.009
- Lindsay N. Childs, Some Hopf Galois structures arising from elementary abelian $p$-groups, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3453–3460. MR 2336557, DOI 10.1090/S0002-9939-07-08888-0
- Lindsay N. Childs and Jesse Corradino, Cayley’s theorem and Hopf Galois structures for semidirect products of cyclic groups, J. Algebra 308 (2007), no. 1, 236–251. MR 2290920, DOI 10.1016/j.jalgebra.2006.09.016
- Daniel Gorenstein, Finite simple groups, University Series in Mathematics, Plenum Publishing Corp., New York, 1982. An introduction to their classification. MR 698782, DOI 10.1007/978-1-4684-8497-7
- Cornelius Greither and Bodo Pareigis, Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), no. 1, 239–258. MR 878476, DOI 10.1016/0021-8693(87)90029-9
- Timothy Kohl, Groups of order $4p$, twisted wreath products and Hopf-Galois theory, J. Algebra 314 (2007), no. 1, 42–74. MR 2331752, DOI 10.1016/j.jalgebra.2007.04.001
Additional Information
- Lindsay N. Childs
- Affiliation: Department of Mathematics and Statistics, University at Albany, Albany, New York 12222
- Email: childs@math.albany.edu
- Received by editor(s): November 18, 2009
- Received by editor(s) in revised form: November 14, 2010, July 21, 2011, and August 25, 2011
- Published electronically: September 21, 2012
- Additional Notes: The author thanks the mathematics department at Virginia Commonwealth University for its hospitality while this research was conducted and the referee for numerous helpful suggestions.
- Communicated by: Ted Chinburg
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1255-1265
- MSC (2000): Primary 12F10
- DOI: https://doi.org/10.1090/S0002-9939-2012-11418-2
- MathSciNet review: 3008873