2014/12/18 by Nigel P. Byott, Byott, Nigel P. · 4 citations
Mathematics · #12F10 #16T05 #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1412.5923
openalex publication_date 2014/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let L/K be a finite Galois extension of fields with group Γ. Associated to each Hopf-Galois structure on L/K is a group G of the same order as the Galois group Γ. The type of the Hopf-Galois structure is by definition the isomorphism type of G. We investigate the extent to which general properties of either of the groups Γ and G constrain those of the other. Specifically, we show that if G is nilpotent then Γ is soluble, and that if Γ is abelian then G is soluble. The proof of the latter result depends on the classification of finite simple groups. In contrast to these results, we give some examples where the groups Γ and G have different composition factors. In particular, we show that a soluble extension may admit a Hopf-Galois structure of insoluble type.