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

Kohl, Timothy

doi:10.2140/ant.2016.10.37

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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

DOI: 10.2140/ant.2016.10.37

Abstract

For $Γ$ a group of order $m p$ , where $p$ is a prime with $gcd (p, m) = 1$ , we consider the regular subgroups $N \leq Perm (Γ)$ 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 $L ∕ K$ with $Γ = Gal (L ∕ K)$ . 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 $m p$ 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 $m p$ .

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

Vol. 10, No. 1, 2016