Open Access
2013 Regular permutation groups of order mp and Hopf Galois structures
Timothy Kohl
Algebra Number Theory 7(9): 2203-2240 (2013). DOI: 10.2140/ant.2013.7.2203


Let Γ be a group of order mp where p is prime and p>m. We give a strategy to enumerate the regular subgroups of Perm(Γ) normalized by the left representation λ(Γ) of Γ. These regular subgroups are in one-to-one correspondence with the Hopf Galois structures on Galois field extensions LK with Γ= Gal(LK). We prove that every such regular subgroup is contained in the normalizer in Perm(Γ) of the p-Sylow subgroup of λ(Γ). This normalizer has an affine representation that makes feasible the explicit determination of regular subgroups in many cases. We illustrate our approach with a number of examples, including the cases of groups whose order is the product of two distinct primes and groups of order p(p1), where p is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.


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Timothy Kohl. "Regular permutation groups of order mp and Hopf Galois structures." Algebra Number Theory 7 (9) 2203 - 2240, 2013.


Received: 8 September 2012; Revised: 2 February 2013; Accepted: 11 March 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1286.12002
MathSciNet: MR3152012
Digital Object Identifier: 10.2140/ant.2013.7.2203

Primary: 20B35
Secondary: 12F10 , 16W30 , 20E22

Keywords: holomorph , Hopf–Galois extension , regular permutation group

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 9 • 2013
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