2018/06/18 by Timothy Kohl, Kohl, Timothy
Mathematics · #16T05 #20B35 #20E07 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1806.06911
openalex publication_date 2018/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Hopf-Galois structures on normal extensions K/k with G=Gal(K/k) are in one-to-one correspondence with the set of regular subgroups N≤ B=Perm(G) that are normalized by the left regular representation λ(G)≤ B. Each such N corresponds to a Hopf algebra HN=(K[N])G that acts on K/k. Such regular subgroups N need not be isomorphic to G but must have the same order. One can subdivide the totality of all such N into collections R(G,[M]) which is the set of those regular N normalized by λ(G) and isomorphic to a given abstract group M where |M|=|G|. There arises an injective correspondence between the characteristic subgroups of a given N an d the set of subgroups of G stemming from the Galois correspondence between sub-Hopf algebras of HN and intermediate fields k⊆ F⊆ K. We utilize this correspondence to show that for certain pairings (G,[M]), the collection R(G,[M]) must be empty.