Hopf-Galois Structures on Non-Normal Extensions of Degree Related to Sophie Germain Primes

We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions \(L/K\) of squarefree degree \(n\). If \(E/K\) is the normal closure of \(L/K\) then \(G=\mathrm{Gal}(E/K)\) can be viewed as a permutation group of degree \(n\). We show that \(G\) has derived length at mo...

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Bibliographic Details
Published in:arXiv.org 2021-06
Main Authors: Byott, Nigel P, Martin-Lyons, Isabel
Format: Article
Language:English
Subjects:
Group theory
Permutations
Online Access:Get full text
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Summary:We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions \(L/K\) of squarefree degree \(n\). If \(E/K\) is the normal closure of \(L/K\) then \(G=\mathrm{Gal}(E/K)\) can be viewed as a permutation group of degree \(n\). We show that \(G\) has derived length at most \(4\), but that many permutation groups of squarefree degree and of derived length \(2\) cannot occur. We then investigate in detail the case where \(n=pq\) where \(q \geq 3\) and \(p=2q+1\) are both prime. (Thus \(q\) is a Sophie Germain prime and \(p\) is a safeprime). We list the permutation groups \(G\) which can arise, and we enumerate the Hopf-Galois structures for each \(G\). There are six such \(G\) for which the corresponding field extensions \(L/K\) admit Hopf-Galois structures of both possible types.
ISSN:2331-8422