Counting Hopf Galois structures on non-abelian Galois field extensions

zbMath0988.12003MaRDI QIDQ1305027

Scott Carnahan, Lindsay N. Childs

Publication date: 4 May 2000

Published in: Journal of Algebra (Search for Journal in Brave)

zbMATH Keywords

symmetric groupsHopf Galois extensionsholomorphs

Mathematics Subject Classification ID

Galois theory (11R32) Separable extensions, Galois theory (12F10) Hopf algebras and their applications (16T05)

Related Items (26)

Normal and non-normal Cayley graphs for symmetric groups ⋮ Induced Hopf Galois structures ⋮ Groups that have the same holomorph as a finite perfect group ⋮ Mutually normalizing regular permutation groups and Zappa-Szép extensions of the holomorph ⋮ Cayley's theorem and Hopf Galois structures for semidirect products of cyclic groups. ⋮ Skew braces and the Galois correspondence for Hopf Galois structures ⋮ Abelian fixed point free endomorphisms and the Yang-Baxter equation ⋮ Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks ⋮ Hopf-Galois structures of squarefree degree ⋮ Hopf-Galois structures on finite extensions with almost simple Galois group ⋮ Hopf-Galois structures on a Galois \(S_n\)-extension ⋮ Rota-Baxter operators on groups ⋮ Hopf-Galois structures on Galois field extensions of degree \(pq\). ⋮ Skew braces and Hopf-Galois structures of Heisenberg type ⋮ Non-existence of Hopf-Galois structures and bijective crossed homomorphisms ⋮ Hopf-Galois structures on almost cyclic field extensions of 2-power degree ⋮ Fixed-point free endomorphisms and Hopf Galois structures ⋮ Characteristic subgroup lattices and Hopf–Galois structures ⋮ Hopf-Galois structures of isomorphic-type on a non-abelian characteristically simple extension ⋮ On \(\mathbb{Z}_n \rtimes \mathbb{Z}_2\)-Hopf-Galois structures ⋮ Abelian Hopf Galois structures on prime-power Galois field extensions ⋮ Finite skew braces with isomorphic non-abelian characteristically simple additive and circle groups ⋮ Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures ⋮ Hopf–Galois structures on finite extensions with quasisimple Galois group ⋮ Enumerating dihedral Hopf-Galois structures acting on dihedral extensions ⋮ Integral Hopf-Galois structures on degree \(p^2\) extensions of \(p\)-adic fields

Uses Software

• GAP

Cites Work

• Classification of the Hopf Galois structures on prime power radical extensions
• Combinatorial structure on the automorphism group of \(S_ 6\)
• Hopf Galois theory for separable field extensions
• Hopf algebras and Galois theory
• Uniqueness of hopf galois structure for separable field extensions
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