Groups of order 4p, twisted wreath products and Hopf–Galois theory

Kohl, Timothy

doi:10.1016/j.jalgebra.2007.04.001

Cited by 20 publications

(15 citation statements)

References 22 publications

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“…Since the computations are in many cases similar to those in the pq case, we provide only a few sample cases to illustrate the variety of approaches needed. This paper generalizes the results for m = 4 in [Kohl 2007]. Some of the ideas here are similar to those in that paper, but for the benefit of the reader we have made this paper independent of [Kohl 2007] and reasonably self-contained.…”

Section: Introductionsupporting

confidence: 65%

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2013

Algebra Number Theory

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We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings with perfect residue fields. We give a number of applications, including a "volume = multiplicity" formula, generalizing the formula of Lazarsfeld and Mustat ,ȃ , and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We prove a generalization of this to generalized symbolic powers of ideals proposed by Herzog, Puthenpurakal and Verma. We also prove an asymptotic "additivity formula" for limits of multiplicities and a formula on limiting growth of valuations, which answers a question posed by the author, Kia Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate filtration on a family of algebraic objects.

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Section: Introductionsupporting

confidence: 65%

“…In [Kohl 2007] we examined groups of order 4 p. There were two cases. If ᏽ = C p × C p = x, y , then there are four linear characters, defined by the following table:…”

Section: Characters and Generators Of P(n)mentioning

confidence: 99%

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2013

Algebra Number Theory

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“…We include here several easily verified properties of the opposite group which can be found in section 3 of [10] for example. (2) is [8,Lemma 2.4.2] where N opp may be constructed explicitly.…”

Section: Regularity and (Multiple) Holomorphsmentioning

confidence: 99%

“…We can go one step further and observe that, since the underlying sets D 2n and Q n are identical, namely Z = t a x b a = 0 1 b = 0 2n − 1 , then we have the following from [10] which we quote verbatim. Proof.…”

Section: Dihedral and Quaternionic Groupsmentioning

confidence: 99%

Multiple Holomorphs of Dihedral and Quaternionic Groups

Kohl

2015

Communications in Algebra

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The holomorph of a group G is Norm B G , the normalizer of the left regular representation G in its group of permutations B = Perm G . The multiple holomorph of G is the normalizer of the holomorph in B. The multiple holomorph and its quotient by the holomorph encodes a great deal of information about the holomorph itself and about the group G and its conjugates within the holomorph. We explore the multiple holomorphs of the dihedral groups D n and quaternionic (dicyclic) groups Q n for n ≥ 3.

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“…Other cases where the classification has been addressed are Galois extensions of order 4p, where p is an odd prime [19], Galois extensions with groups G that are semidirect products of cyclic groups and have trivial centers [13] or Galois extensions or order mp, where p is prime and m < p [20]. A nonunicity result for abelian extensions is given in [6], namely that every finite Galois field extension with abelian group of even order > 4 admits a Hopf Galois structure for which the associated group N is non-abelian.…”

Section: Counting Hopf Galois Structuresmentioning

confidence: 99%

From Galois to Hopf Galois: Theory and Practice

Vicente¹,

Doval²,

Vela³

2015

Contemporary Mathematics

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Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations and explicit descriptions of all the ingredients involved in a Hopf Galois structure. At the end we give just a glimpse of how this theory is used in the context of Galois module theory for wildly ramified extensions.

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Groups of order 4p, twisted wreath products and Hopf–Galois theory

Cited by 20 publications

References 22 publications

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Multiple Holomorphs of Dihedral and Quaternionic Groups

From Galois to Hopf Galois: Theory and Practice

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