Fixed-point free endomorphisms and Hopf Galois structures

Author:
Lindsay N. Childs

Journal:
Proc. Amer. Math. Soc. **141** (2013), 1255-1265

MSC (2000):
Primary 12F10

DOI:
https://doi.org/10.1090/S0002-9939-2012-11418-2

Published electronically:
September 21, 2012

MathSciNet review:
3008873

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a Galois extension of fields with finite Galois group . Greither and Pareigis showed that there is a bijection between Hopf Galois structures on and regular subgroups of normalized by , and Byott translated the problem into that of finding equivalence classes of embeddings of in the holomorph of groups of the same cardinality as . In 2007 we showed, using Byott's translation, that fixed point free endomorphisms of yield Hopf Galois structures on . 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 , the -Hopf algebra that arises from the left regular representation of in . The paper concludes with various old and new examples of abelian fixed point free endomorphisms.

**[By96]**N. P. Byott, Uniqueness of Hopf Galois structure of separable field extensions, Comm. Algebra 24 (1996), 3217-3228. MR**1402555 (97j:16051a)****[By02]**N. P. Byott, Integral Hopf-Galois structures on degree extensions of -adic fields,

J. Algebra 248 (2002), 334-365. MR**1879021 (2002j:11142)****[By04]**N. P. Byott, Hopf-Galois structures on field extensions with simple Galois groups, Bulletin of the London Mathematical Society 36 (2004), 23-29. MR**2011974 (2004i:16049)****[CaC99]**S. Carnahan, L. N. Childs, Counting Hopf Galois structures on non-abelian Galois field extensions, J. Algebra 218 (1999), 81-92. MR**1704676 (2000e:12010)****[Ch00]**L. N. Childs,*Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory*, Mathematical Surveys and Monographs**80**, American Mathematical Society, 2000. MR**1767499 (2001e:11116)****[Ch03]**L. N. Childs, On Hopf Galois structures and complete groups, New York J. Math. 9 (2003), 99-115. MR**2016184 (2004k:16097)****[Ch05]**L. N. Childs, Elementary abelian Hopf Galois structures and polynomial formal groups, J. Algebra 283 (2005), 292-316. MR**2102084 (2005g:16073)****[Ch07]**L. N. Childs, Some Hopf Galois structures arising from elementary abelian -groups, Proc. Amer. Math. Soc. 135 (2007), 3453-3460. MR**2336557 (2008j:16107)****[CCo07]**L. N. Childs, J. Corradino, Cayley's Theorem and Hopf Galois structures for semidirect products of cyclic groups, J. Algebra 308 (2007), 236-251. MR**2290920 (2007j:20026)****[Go82]**D. Gorenstein,*Finite Simple Groups, An Introduction to Their Classification*, Plenum, New York/London, 1982. MR**698782 (84j:20002)****[GP87]**C. Greither, B. Pareigis, Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), 239-258. MR**878476 (88i:12006)****[Ko07]**T. Kohl, Groups of order , twisted wreath products and Hopf-Galois theory, J. Algebra 314 (2007), 42-74. MR**2331752 (2008e:12001)**

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Additional Information

**Lindsay N. Childs**

Affiliation:
Department of Mathematics and Statistics, University at Albany, Albany, New York 12222

Email:
childs@math.albany.edu

DOI:
https://doi.org/10.1090/S0002-9939-2012-11418-2

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

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.