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2016 Hopf–Galois structures arising from groups with unique subgroup of order $p$
Timothy Kohl
Algebra Number Theory 10(1): 37-59 (2016). DOI: 10.2140/ant.2016.10.37

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.

Citation

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Timothy Kohl. "Hopf–Galois structures arising from groups with unique subgroup of order $p$." Algebra Number Theory 10 (1) 37 - 59, 2016. https://doi.org/10.2140/ant.2016.10.37

Information

Received: 5 August 2014; Revised: 1 October 2015; Accepted: 27 November 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1341.20002
MathSciNet: MR3463035
Digital Object Identifier: 10.2140/ant.2016.10.37

Subjects:
Primary: 20B35
Secondary: 16T05, 20D20, 20D45

Rights: Copyright © 2016 Mathematical Sciences Publishers

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Vol.10 • No. 1 • 2016
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