On the Galois correspondence for Hopf Galois structures arising from finite radical algebras and Zappa-Szép products

Abstract

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-Szép products of finite groups, and in particular when A 3 = 0 or the Zappa-Szép 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, •) ∼ = Z 2m is cyclic and (G, ) ∼ = Dm is dihedral, then for large m, GC(Z 2m , Dm) is close to 1/2 while GC(Dm, Z 2m ) is near 0.

On the connection between Hopf–Galois structures and skew braces

“…We study now a question posed in [16]. Let $$L_1/K_1$$ be a finite Galois extension of fields with Galois group $$(G,\circ )$$, and consider a Hopf–Galois structure on $$L_1/K_1$$, with associated skew brace $$(G,\cdot ,\circ )$$.…”

“…We study now a question posed in [16]. Let 𝐿 1 ∕𝐾 1 be a finite Galois extension of fields with Galois group (𝐺, •), and consider a Hopf-Galois structure on 𝐿 1 ∕𝐾 1 , with associated skew brace (𝐺, ⋅, •).…”

“…Despite this promising start, to the best knowledge of the authors, no further examples of this behaviour have been found. A new approach, introduced in [16], seems to suggest how difficult it is to find them. Namely, instead of looking for Hopf-Galois structures for which the Hopf-Galois correspondence is surjective, one can study the failure of the surjectivity.…”