 

Nigel P. Byott and Lindsay N. Childs
Fixedpoint free pairs of homomorphisms and nonabelian HopfGalois structures view print


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


Abstract
Given finite groups Γ and G of order n, regular embeddings
from Γ to the holomorph of G yield HopfGalois structures on a
Galois extension LK of fields with Galois group Γ. Here we
consider regular embeddings that arise from fixedpoint free pairs of
homomorphisms from Γ to G. If G is a complete group, then all
regular embeddings arise from fixedpoint free pairs. For all Γ,
G of order n = p(p1) with p a safeprime, we compute the number
of HopfGalois structures that arise from fixedpoint free pairs, and
compare the results with a count of all HopfGalois structures
obtained by T. Kohl. Using the idea of fixedpoint free pairs, we
characterize the abelian Galois groups Γ of even order or order a
power of p, an odd prime, for which LK admits a nonabelian Hopf
Galois structure. The paper concludes with some new classes of
abelian groups Γ for which every HopfGalois 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
N.P.Byott@ex.ac.uk
Lindsay N. Childs:
Department of Mathematics and Statistics, University at Albany, Albany, NY 12222
lchilds@albany.edu

