Hopf–Galois structures arising from groups with unique subgroup of order p
| Autor: | Timothy Kohl |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Mathematics::Number Theory
Regular representation Group Theory (math.GR) 01 natural sciences Prime (order theory) Combinatorics 16T05 Mathematics::Group Theory 20B35 20D20 20D45 16T05 0103 physical sciences FOS: Mathematics Order (group theory) 20D45 0101 mathematics regular subgroup 20D20 Mathematics Algebra and Number Theory Group (mathematics) 010102 general mathematics Sylow theorems Hopf–Galois extension Mathematics - Rings and Algebras Automorphism Centralizer and normalizer Rings and Algebras (math.RA) Field extension 20B35 010307 mathematical physics Mathematics - Group Theory |
| Zdroj: | Algebra Number Theory 10, no. 1 (2016), 37-59 |
| ISSN: | 1944-7833 1937-0652 |
| DOI: | 10.2140/ant.2016.10.37 |
| Popis: | For $\Gamma$ a group of order $mp$ for $p$ prime where $gcd(p,m)=1$, we consider those regular subgroups $N\leq Perm(\Gamma)$ normalized by $\lambda(\Gamma)$, the left regular representation of $\Gamma$. These subgroups are in one-to-one correspondence with the Hopf-Galois structures on separable field extensions $L/K$ with $\Gamma=Gal(L/K)$. This is a follow up to the author's earlier work where, by assuming $p>m$, one has that all such $N$ lie within the normalizer of the $p$-Sylow subgroup of $\lambda(\Gamma)$. Here we show that one only need assume that all groups of a given order $mp$ have a unique $p$-Sylow subgroup, and that $p$ not be a divisor of the automorphism groups of any group of order $m$. As such, we extend the applicability of the program for computing these regular subgroups $N$ and concordantly the corresponding Hopf-Galois structures on separable extensions of degree $mp$. |
| Databáze: | OpenAIRE |
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