| Abstract: |
Let L/K be a G-Galois extension of fields with an H-Hopf Galois structure of type N. We study the Galois correspondence ratio GC(G, N), which is the proportion of intermediate fields E with K ⊆ E ⊆ L that are in the image of the Galois correspondence for the H-Hopf Galois structure on L/K. The Galois correspondence ratio for a Hopf Galois structure can be found by translating the problem into counting certain subgroups of the corresponding skew brace. We look at skew braces arising from finite radical algebras A and from Zappa-Sz'ep products of finite groups, and in particular when A3 = 0 or the Zappa-Sz'ep product is a semidirect product, in which cases the corresponding skew brace is a bi-skew brace, that is, a set G with two group operations ◦ and ? in such a way that G is a skew brace with either group structure acting as the additive group of the skew brace. We obtain the Galois correspondence ratio for several examples. In particular, if (G, ◦, ?) is a biskew brace of squarefree order 2m where (G, ◦) ∼= Z2m is cyclic and (G, ?) ∼= Dm is dihedral, then for large m, GC(Z2m, Dm) is close to 1/2 while GC(Dm, Z2m) is near 0. |
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