Algebra & Number Theory

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

Timothy Kohl

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 $L∕K$ 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 $mp$ 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

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

Dates
Revised: 1 October 2015
Accepted: 27 November 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842464

Digital Object Identifier
doi:10.2140/ant.2016.10.37

Mathematical Reviews number (MathSciNet)
MR3463035

Zentralblatt MATH identifier
1341.20002

Citation

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. https://projecteuclid.org/euclid.ant/1510842464

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