Algebra & Number Theory

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

Timothy Kohl

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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.

Article information

Algebra Number Theory, Volume 10, Number 1 (2016), 37-59.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20B35: Subgroups of symmetric groups
Secondary: 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure 20D45: Automorphisms 16T05: Hopf algebras and their applications [See also 16S40, 57T05]

Hopf–Galois extension regular subgroup


Kohl, Timothy. Hopf–Galois structures arising from groups with unique subgroup of order $p$. Algebra Number Theory 10 (2016), no. 1, 37--59. doi:10.2140/ant.2016.10.37.

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