Vol. 7, No. 9, 2013

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Regular permutation groups of order mp and Hopf Galois structures

Timothy Kohl

Vol. 7 (2013), No. 9, 2203–2240

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(p 1), where p is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.

regular permutation group, Hopf–Galois extension, holomorph
Mathematical Subject Classification 2010
Primary: 20B35
Secondary: 12F10, 20E22, 16W30
Received: 8 September 2012
Revised: 2 February 2013
Accepted: 11 March 2013
Published: 18 December 2013
Timothy Kohl
Department of Mathematics and Statistics
Boston University
111 Cummington Mall
Boston, MA 02215
United States