2019/07/17 by Lindsay N. Childs, Childs, Lindsay N.
Mathematics · #12F10 (primary) #16T05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1907.07711
openalex publication_date 2019/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let L/K be a G-Galois extension of fields with an H-Hopf Galois\nstructure of type N. We study the ratio GC(G, N), which is the number of\nintermediate fields E with K \⊆ E \⊆ L that are in the image\nof the Galois correspondence for the H-Hopf Galois structure on L/K,\ndivided by the number of intermediate fields. By Galois descent, L \⊗K H\n= LN where N is a G-invariant regular subgroup of \Perm(G), and\nthen GC(G, N) is the number of G-invariant subgroups of N, divided by the\nnumber of subgroups of G. We look at the Galois correspondence ratio for a\nHopf Galois structure by translating the problem into counting certain\nsubgroups of the corresponding skew brace. We look at skew braces arising from\nfinite radical algebras A and from Zappa-Sz 'ep products of finite groups,\nand in particular when A3 = 0 or the Zappa-Sz 'ep product is a semidirect\nproduct, in which cases the corresponding skew brace is a bi-skew brace, that\nis, a set G with two group operations \∘ and \⋆ in such a way that\nG is a skew brace with either group structure acting as the additive group of\nthe skew brace. We obtain the Galois correspondence ratio for several examples.\nIn particular, if (G, \∘, \⋆) is a bi-skew brace of squarefree order\n2m where (G, \∘) \≅ Z2m is cyclic and (G, \⋆) = Dm is\ndihedral, then for large m, GC(Z2m,Dm), is close to 1/2 while GC(Dm,\nZ2m) is near 0.\n