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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
Published electronically: September 21, 2012
MathSciNet review: 3008873
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Abstract: Let $ L\vert 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\vert 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\vert 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.

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Additional Information

Lindsay N. Childs
Affiliation: Department of Mathematics and Statistics, University at Albany, Albany, New York 12222

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.

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