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Hopf–Galois structures arising from groups with unique subgroup of order p
- Source :
- Algebra Number Theory 10, no. 1 (2016), 37-59
- Publication Year :
- 2016
- Publisher :
- Mathematical Sciences Publishers, 2016.
-
Abstract
- 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 $mp$.
- Subjects :
- 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
Subjects
Details
- ISSN :
- 19447833 and 19370652
- Volume :
- 10
- Database :
- OpenAIRE
- Journal :
- Algebra & Number Theory
- Accession number :
- edsair.doi.dedup.....8be7266dae73762517ea26bdca9f6dd0