Regular permutation groups of order mp and Hopf Galois structures

Kohl, Timothy

doi:10.2140/ant.2013.7.2203

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Regular permutation groups of order mp and Hopf Galois structures

Timothy Kohl

Vol. 7 (2013), No. 9, 2203–2240

DOI: 10.2140/ant.2013.7.2203

Abstract

Let $Γ$ be a group of order $m p$ 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.

Keywords

regular permutation group, Hopf–Galois extension, holomorph

Mathematical Subject Classification 2010

Primary: 20B35

Secondary: 12F10, 20E22, 16W30

Milestones

Received: 8 September 2012

Revised: 2 February 2013

Accepted: 11 March 2013

Published: 18 December 2013

Authors

Timothy Kohl
Department of Mathematics and Statistics Boston University 111 Cummington Mall Boston, MA 02215 United States

Vol. 7, No. 9, 2013