Algebra & Number Theory

Regular permutation groups of order mp and Hopf Galois structures

Timothy Kohl

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Abstract

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 L∕K with Γ= Gal(L∕K). 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(p−1), where p is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.

Article information

Source
Algebra Number Theory, Volume 7, Number 9 (2013), 2203-2240.

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730091

Digital Object Identifier
doi:10.2140/ant.2013.7.2203

Mathematical Reviews number (MathSciNet)
MR3152012

Zentralblatt MATH identifier
1286.12002

Subjects
Primary: 20B35: Subgroups of symmetric groups
Secondary: 12F10: Separable extensions, Galois theory 20E22: Extensions, wreath products, and other compositions [See also 20J05] 16W30

Keywords
regular permutation group Hopf–Galois extension holomorph

Citation

Kohl, Timothy. Regular permutation groups of order mp and Hopf Galois structures. Algebra Number Theory 7 (2013), no. 9, 2203--2240. doi:10.2140/ant.2013.7.2203. https://projecteuclid.org/euclid.ant/1513730091


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