New York Journal of Mathematics
Volume 18 (2012) 707-731


Nigel P. Byott and Lindsay N. Childs

Fixed-point free pairs of homomorphisms and nonabelian Hopf-Galois structures

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Published: October 6, 2012
Keywords: Hopf-Galois structure; abelian extensions; semidirect product
Subject: 12F10 (primary), 16W30 (secondary)

Given finite groups Γ and G of order n, regular embeddings from Γ to the holomorph of G yield Hopf-Galois structures on a Galois extension L|K of fields with Galois group Γ. Here we consider regular embeddings that arise from fixed-point free pairs of homomorphisms from Γ to G. If G is a complete group, then all regular embeddings arise from fixed-point free pairs. For all Γ, G of order n = p(p-1) with p a safeprime, we compute the number of Hopf-Galois structures that arise from fixed-point free pairs, and compare the results with a count of all Hopf-Galois structures obtained by T. Kohl. Using the idea of fixed-point free pairs, we characterize the abelian Galois groups Γ of even order or order a power of p, an odd prime, for which L|K admits a nonabelian Hopf Galois structure. The paper concludes with some new classes of abelian groups Γ for which every Hopf-Galois structure has type Γ (and hence is abelian).

Author information

Nigel P. Byott:
College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, UK

Lindsay N. Childs:
Department of Mathematics and Statistics, University at Albany, Albany, NY 12222