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

Abstract

Let $L/K$ be a $G$-Galois extension of fields with an $H$-Hopf Galois structure of type $N$. We study the ratio $GC(G,N)$, which is the number of intermediate fields $E$ with $K\subseteq E\subseteq L$ that are in the image of the Galois correspondence for the $H$-Hopf Galois structure on $L/K$, divided by the number of intermediate fields. By Galois descent, $L{\otimes }_{K}H=LN$ where $N$ is a $G$-invariant regular subgroup of $\mathrm{Perm}(G)$, and then $GC(G,N)$ is the number of $G$-invariant subgroups of $N$, divided by the number of subgroups of $G$. We look at the Galois correspondence ratio for a Hopf Galois structure 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 $\circ$ and $\star$ 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 $(G,\circ ,\star )$ is a bi-skew brace of squarefree order $2m$ where $(G,\circ )\cong Z_{2m}$ is cyclic and $(G,\star )=D_m$ is dihedral, then for large $m$, $GC(Z_{2m},D_m),$ is close to 1/2 while $GC(D_m,Z_{2m})$ is near 0.

23 pages. Some computations in the examples were corrected. The final dihedral example was generalized. Submitted to Publ. Mat. (Barcelona)