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

Lindsay N. Childs
2021 Publicacions matemàtiques  
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
more » ... 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. 2010 Mathematics Subject Classification: 12F10.
doi:10.5565/publmat6512105 fatcat:eoz7aefngnfm5omentyx4vw2re