Vol. 10, No. 1, 2016

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Hopf–Galois structures arising from groups with unique subgroup of order $p$

Timothy Kohl

Vol. 10 (2016), No. 1, 37–59
Abstract

For Γ a group of order mp, where p is a prime with gcd(p,m) = 1, we consider the regular subgroups N Perm(Γ) that are normalized by λ(Γ), the left regular representation of Γ. These subgroups are in one-to-one correspondence with the Hopf–Galois structures on separable field extensions LK with Γ = Gal(LK). Elsewhere we showed that if p > m then all such N lie within the normalizer of the Sylow p-subgroup of λ(Γ). Here we show that one only need assume that all groups of a given order mp have a unique Sylow p-subgroup, and that p not be a divisor of the order of the automorphism groups of any group of order m. We thus extend the applicability of the program for computing these regular subgroups N and concordantly the corresponding Hopf–Galois structures on separable extensions of degree mp.

Keywords
Hopf–Galois extension, regular subgroup
Mathematical Subject Classification 2010
Primary: 20B35
Secondary: 20D20, 20D45, 16T05
Milestones
Received: 5 August 2014
Revised: 1 October 2015
Accepted: 27 November 2015
Published: 14 February 2016
Authors
Timothy Kohl
Department of Mathematics and Statistics
Boston University
111 Cummington Mall
Boston, MA 02215
United States