On the Galois correspondence for Hopf Galois structures arising from finite radical algebras and Zappa-Sz\'ep products

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-Sz\'ep products of finite groups, and in particular when $A^3 = 0$ 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 $\circ$ 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.Comment: 23 pages. Some computations in the examples were corrected. The final dihedral example was generalized. Submitted to Publ. Mat. (Barcelona

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

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

Scaffolds and Generalized Integral Galois Module Structure

Let $L/K$ be a finite, totally ramified $p$-extension of complete local fields with residue fields of characteristic $p > 0$, and let $A$ be a $K$-algebra acting on $L$. We define the concept of an $A$-scaffold on $L$, thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where $L/K$ was Galois and $A=K[G]$ for $G=\mathrm{Gal}(L/K)$. When a suitable $A$-scaffold exists, we show how to answer questions generalizing those of classical integral Galois module theory. We give a necessary and sufficient condition, involving only numerical parameters, for a given fractional ideal to be free over its associated order in $A$. We also show how to determine the number of generators required when it is not free, along with the embedding dimension of the associated order. In the Galois case, the numerical parameters are the ramification breaks associated with $L/K$. We apply these results to biquadratic Galois extensions in characteristic 2, and to totally and weakly ramified Galois $p$-extensions in characteristic $p$. We also apply our results to the non-classical situation where $L/K$ is a finite primitive purely inseparable extension of arbitrary exponent that is acted on, via a higher derivation (but in many different ways), by the divided power $K$-Hopf algebra.Comment: Further minor corrections and improvements to exposition. Reference [BE] updated. To appear in Ann. Inst. Fourier, Grenobl

Fixed-point free pairs of homomorphisms and nonabelian Hopf-Galois structure

Given finite groups Γ and G of order n, regular embeddings from Γ to the holomorph of G yield Hopf-Galois structures on a Galois extension L|K of fields with Galois group Γ. Here we consider regular embeddings that arise from fixed-point free pairs of homomorphisms from Γ to G. If G is a complete group, then all regular embeddings arise from fixed-point free pairs. For all Γ, G of order n = p(p-1) with p a safeprime, we compute the number of Hopf-Galois structures that arise from fixed-point free pairs, and compare the results with a count of all Hopf-Galois structures obtained by T. Kohl. Using the idea of fixed-point free pairs, we characterize the abelian Galois groups Γ of even order or order a power of p, an odd prime, for which L|K admits a nonabelian Hopf Galois structure. The paper concludes with some new classes of abelian groups Γ for which every Hopf-Galois structure has type Γ (and hence is abelian)

Fixed point free automorphisms of groups related to finite fields

AbstractLet G=Fq⋊〈β〉 be the semidirect product of the additive group of the field of q=pn elements and the cyclic group of order d generated by the invertible linear transformation β defined by multiplication by a power of a primitive root of Fq. We find an arithmetic condition on d so that every endomorphism of G is determined by its values on (1,1) and (0,β). When that is the case, we determine the fixed point free automorphisms of G. If d equals the odd part of q−1 then we count the fixed point free automorphisms of G—such exist only when p is a Fermat prime