Hopf Galois Structures Research Articles

Minimal Hopf-Galois Structures on Separable Field Extensions

In Hopf-Galois theory, every -Hopf-Galois structure on a field extension gives rise to an injective map from the set of -sub-Hopf algebras of into the intermediate fields of . Recent papers on the failure of the surjectivity of reveal that there exist many Hopf-Galois structures for which there are many more subfields than sub-Hopf algebras. In this paper we survey and illustrate group-theoretical methods to determine -Hopf-Galois structures on finite separable extensions in the extreme situation when has only two sub-Hopf algebras. This corresponds to the case when the lack of surjectivity is at its extreme.

Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures

Let G G be a finite nonabelian group, and let ψ : G → G \psi :G\to G be a homomorphism with abelian image. We show how ψ \psi gives rise to two Hopf-Galois structures on a Galois extension L / K L/K with Galois group (isomorphic to) G G ; one of these structures generalizes the construction given by a “fixed point free abelian endomorphism” introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.

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.

Automatic realization of Hopf Galois structures

We consider Hopf Galois structures on a separable field extension [Formula: see text] of degree [Formula: see text], for [Formula: see text] an odd prime number, [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] has at most one abelian type of Hopf Galois structures. For a nonabelian group [Formula: see text] of order [Formula: see text], with commutator subgroup of order [Formula: see text], we prove that if [Formula: see text] has a Hopf Galois structure of type [Formula: see text], then it has a Hopf Galois structure of type [Formula: see text], where [Formula: see text] is an abelian group of order [Formula: see text] and having the same number of elements of order [Formula: see text] as [Formula: see text], for [Formula: see text].

Hopf Galois structures on field extensions of degree twice an odd prime square and their associated skew left braces

We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding skew left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.

Hopf–Galois structures on finite extensions with quasisimple Galois group

Let $L/K$ be a finite Galois extension of fields with Galois group $G$. It is known that $L/K$ admits exactly two Hopf-Galois structures when $G$ is non-abelian simple. In this paper, we extend this result to the case when $G$ is quasisimple.

Skew braces of squarefree order

Let [Formula: see text] be a squarefree integer, and let [Formula: see text], [Formula: see text] be two groups of order [Formula: see text]. Using our previous results on the enumeration of Hopf–Galois structures on Galois extensions of fields of squarefree degree, we determine the number of skew braces (up to isomorphism) with multiplicative group [Formula: see text] and additive group [Formula: see text]. As an application, we enumerate skew braces whose order is the product of three distinct primes, in particular proving a conjecture of Bardakov, Neshchadim and Yadav on the number of skew braces of order [Formula: see text] for primes [Formula: see text].

Hopf-Galois structures on finite extensions with almost simple Galois group

In this paper, we study the Hopf-Galois structures on a finite Galois extension whose Galois group G is an almost simple group in which the socle A has prime index p. Each Hopf-Galois structure is associated to a group N of the same order as G. We shall give necessary criteria on these N in terms of their group-theoretic properties, and determine the number of Hopf-Galois structures associated to A×Cp, where Cp is the cyclic group of order p.

Hopf-Galois structures of squarefree degree

Let n be a squarefree natural number, and let G, Γ be two groups of order n. We determine the number of Hopf-Galois structures of type G admitted by a Galois extension of fields with Galois group isomorphic to Γ. We give some examples, including a full treatment of the case where n is the product of three primes.

Hopf-Galois structures on extensions of degree p2q and skew braces of order p2q: The cyclic Sylow p-subgroup case

Let p,q be distinct primes, with p>2.We classify the Hopf-Galois structures on Galois extensions of degree p2q, such that the Sylow p-subgroups of the Galois group are cyclic.This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups (G,⋅) of order p2q, in the case when the Sylow p-subgroups of G are cyclic. This is equivalent to classifying the skew braces (G,⋅,∘).Furthermore, we prove that if G and Γ are groups of order p2q with non-isomorphic Sylow p-subgroups, then there are no regular subgroups of the holomorph of G which are isomorphic to Γ. Equivalently, a Galois extension with Galois group Γ has no Hopf-Galois structures of type G.Our method relies on the alternate brace operation ∘ on G, which we use mainly indirectly, that is, in terms of the functions γ:G→Aut(G) defined by g↦(x↦(x∘g)⋅g−1). These functions are in one-to-one correspondence with the regular subgroups of the holomorph of G, and are characterised by the functional equation γ(gγ(h)⋅h)=γ(g)γ(h), for g,h∈G. We develop methods to deal with these functions, with the aim of making their enumeration easier, and more conceptual.

Computation of Hopf Galois structures on separable extensions and classification of those for degree twice an odd prime power

A Hopf Galois structure on a finite field extension [Formula: see text] is a pair [Formula: see text], where [Formula: see text] is a finite cocommutative [Formula: see text]-Hopf algebra and [Formula: see text] a Hopf action. In this paper, we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree [Formula: see text], with [Formula: see text] an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the occurrence of some other type. In particular, for separable field extensions of degree [Formula: see text], we determine exactly the possible sets of Hopf Galois structure types.

Computation of Hopf Galois structures on low degree separable extensions and classification of those for degrees $p^2$ and $2p$

A Hopf Galois structure on a finite field extension L/K is a pair (H,µ), where H is a finite cocommutative K-Hopf algebra and µ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of degree up to eleven and several properties of those. Besides, we exhibit several results on Hopf Galois structures inspired by the program output. We prove that if (H,µ) is an almost classically Hopf Galois structure, then it is the unique Hopf Galois structure with underlying Hopf algebra H up to isomorphism. For p an odd prime, we prove that a separable extension of degree p2 may have only one type of Hopf Galois structure and determine those of cyclic type; we determine as well the Hopf Galois structures on separable extensions of degree 2p. We highlight the richness of the results obtained for extensions of degree 8 by computing an explicit example and presenting some tables which summarize these results.

Opposite skew left braces and applications

Given a skew left brace B, we introduce the notion of an “opposite” skew left brace B′, which is closely related to the concept of the opposite of a group, and provide several applications. Skew left braces are closely linked with both solutions to the Yang-Baxter Equation and Hopf-Galois structures on Galois field extensions. We show that the set-theoretic solution to the YBE given by B′ is the inverse to the solution given by B. Every Hopf-Galois structure on a Galois field extension L/K gives rise to a skew left brace B; if the underlying Hopf algebra is not commutative, then one can construct an additional “opposite” Hopf-Galois structure (see [1], which relates the Hopf-Galois module structures of each, and refers to the structures as “commuting”); the corresponding skew left brace to this second structure is precisely B′. We show how left ideals (and a newly introduced family of quasi-ideals) of B′ allow us to identify the intermediate fields of L/K which occur as fixed fields of sub-Hopf algebras under this correspondence and to identify which of these are Galois, or Hopf-Galois, over K. Finally, we use the opposite to connect the inverse solution to the YBE and the structure of the Hopf algebra H acting on L/K; this allows us to identify the group-like elements of H.

Enumerating Dihedral Hopf-Galois structures acting on Dihedral extensions

The work of Greither and Pareigis details the enumeration of the Hopf-Galois structures (if any) on a given separable field extension. For an extension L/K which is classically Galois with G=Gal(L/K) the Hopf algebras in question are of the form (L[N])G where N≤B=Perm(G) is a regular subgroup that is normalized by the left regular representation λ(G)≤B. We consider the case where both G and N are isomorphic to a dihedral group Dn for any n≥3. Using the normal block systems inherent to the left regular representation of each Dn, (and every other regular permutation group isomorphic to Dn) we explicitly enumerate all possible such N which arise.

Hopf-Galois module structure of tamely ramified radical extensions of prime degree

Let K be a number field and let L/K be a tamely ramified radical extension of prime degree p. If K contains a primitive pth root of unity then L/K is a cyclic Kummer extension; in this case the group algebra K[G] (with G=Gal(L/K)) gives the unique Hopf-Galois structure on L/K, the ring of algebraic integers OL is locally free over OK[G] by Noether's theorem, and Gómez Ayala has determined a criterion for OL to be a free OK[G]-module. If K does not contain a primitive pth root of unity then L/K is a separable, but non-normal, extension, which again admits a unique Hopf-Galois structure. Under the assumption that p is unramified in K, we show that OL is locally free over its associated order in this Hopf-Galois structure and determine a criterion for it to be free. We find that the conditions that appear in this criterion are identical to those appearing in Gómez Ayala's criterion for the normal case.

Hopf-Galois structures of isomorphic-type on a non-abelian characteristically simple extension

Let L / K L/K be a finite Galois extension whose Galois group G G is non-abelian and characteristically simple. By drawing tools from graph theory and group theory, we shall give a closed formula for the total number of Hopf-Galois structures on L / K L/K with associated group isomorphic to G G .

Hopf-Galois structures on a Galois Sn-extension

In this paper, we shall determine the exact number of Hopf-Galois structures on a Galois Sn-extension, where Sn denotes the symmetric group on n letters.

Characteristic subgroup lattices and Hopf–Galois structures

The Hopf–Galois structures on normal field extensions [Formula: see text] with [Formula: see text] are in one-to-one correspondence with the set of regular subgroups [Formula: see text] of [Formula: see text], the group of permutations of [Formula: see text] as a set, that are normalized by the left regular representation [Formula: see text]. Each such [Formula: see text] corresponds to a Hopf algebra [Formula: see text] that acts on [Formula: see text]. Such regular subgroups need not be isomorphic to [Formula: see text] but must have the same order. One can divide all such [Formula: see text] into collections [Formula: see text], where [Formula: see text] is the set of those regular [Formula: see text] normalized by [Formula: see text] and isomorphic to a given abstract group [Formula: see text], where [Formula: see text]. There exists an injective correspondence between the characteristic subgroups of a given [Formula: see text] and the set of subgroups of [Formula: see text] stemming from the Galois correspondence between sub-Hopf algebras of [Formula: see text] and intermediate fields [Formula: see text], where [Formula: see text]. We utilize this correspondence to show that for certain pairs [Formula: see text], the collection [Formula: see text] must be empty. This also shows that for these [Formula: see text], there do not exist skew braces with additive group isomorphic to [Formula: see text] and circle group isomorphic to [Formula: see text].

Normality and short exact sequences of Hopf-Galois structures

Every Hopf-Galois structure on a finite Galois extension K/k where corresponds uniquely to a regular subgroup , normalized by , in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra which acts on K/k is . For a given such N we consider the Hopf-Galois structure arising from a subgroup that is also normalized by . This subgroup gives rise to a Hopf sub-algebra with fixed field . By the work of Chase and Sweedler, this yields a Hopf-Galois structure on the extension K/F where the action arises by base changing HP to which is an F-Hopf algebra. We examine this analogy with classical Galois theory, and also examine how the Hopf-Galois structure on K/F relates to that on K/k. We will also pay particular attention to how the Greither-Pareigis enumeration/construction of those HP acting on K/F relates to that of the HN which act on K/k. In the process we also examine short exact sequences of the Hopf algebras which act, whose exactness is directly tied to the descent theoretic description of these algebras.

Skew braces and Hopf–Galois structures of Heisenberg type

We classify all skew braces of Heisenberg type for a prime number p>3. Furthermore, we determine the automorphism group of each one of these skew braces (as well as their socle and annihilator). Hence, by utilising a link between skew braces and Hopf–Galois theory, we can determine all Hopf–Galois structures of Heisenberg type on Galois field extensions of fields of degree p3.