scholarly journals Automatic realization of Hopf Galois structures

2020 ◽  
pp. 2250030
Author(s):  
Teresa Crespo
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Commutator Subgroup ◽  
Field Extension ◽  
Nonabelian Group ◽  
Type Formula ◽  
Galois Structure ◽  
Number Formula

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].

scholarly journals On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions

2018 ◽  
Vol 2018 (734) ◽  
pp. 187-228
Author(s):  
David Burns ◽  
Daniel Macias Castillo ◽  
Christian Wuthrich
Keyword(s):  
Prime Number ◽  
Abelian Variety ◽  
Galois Extension ◽  
Field Extension ◽  
Galois Structure ◽  
Weil Group ◽  
Relevant Field ◽  
Tamagawa Number Conjecture ◽  
Derivatives Of ◽  
Positive Rank

AbstractLetAbe an abelian variety defined over a number fieldkand letFbe a finite Galois extension ofk. Letpbe a prime number. Then under certain not-too-stringent conditions onAandFwe compute explicitly the algebraic part of thep-component of the equivariant Tamagawa number of the pair(h^{1}(A_{/F})(1),\mathbb{Z}[{\rm Gal}(F/k)]). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of thep-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell–Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible byp. More generally, our approach leads us to the formulation of certain precise families of conjecturalp-adic congruences between the values ats=1of derivatives of the Hasse–WeilL-functions associated to twists ofA, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate–Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions.

scholarly journals ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

WARFIELD INVARIANTS IN COMMUTATIVE GROUP ALGEBRAS

2008 ◽  
Vol 07 (03) ◽  
pp. 337-346 ◽  
Author(s):  
PETER V. DANCHEV
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Group Algebra ◽  
Ordinal Number ◽  
Commutative Group ◽  
Group Algebras

Let F be a field and G an Abelian group. For every prime number q and every ordinal number α we compute only in terms of F and G the Warfield q-invariants Wα, q(VF[G]) of the group VF[G] of all normed units in the group algebra F[G] under some minimal restrictions on F and G. This expands own recent results from (Extracta Mathematicae, 2005) and (Collectanea Mathematicae, 2008).

On a conjecture about the quasirecognition by prime graph of some finite simple groups

2019 ◽  
Vol 18 (04) ◽  
pp. 1950070
Author(s):  
Ali Mahmoudifar
Keyword(s):  
Computer Program ◽  
Natural Number ◽  
Simple Group ◽  
Prime Number ◽  
Prime Power ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Number Formula ◽  
Finite Linear

It is proved that some finite simple groups are quasirecognizable by prime graph. In [A. Mahmoudifar and B. Khosravi, On quasirecognition by prime graph of the simple groups [Formula: see text] and [Formula: see text], J. Algebra Appl. 14(1) (2015) 12pp], the authors proved that if [Formula: see text] is a prime number and [Formula: see text], then there exists a natural number [Formula: see text] such that for all [Formula: see text], the simple group [Formula: see text] (where [Formula: see text] is a linear or unitary simple group) is quasirecognizable by prime graph. Also[Formula: see text] in that paper[Formula: see text] the author posed the following conjecture: Conjecture. For every prime power [Formula: see text] there exists a natural number [Formula: see text] such that for all [Formula: see text] the simple group [Formula: see text] is quasirecognizable by prime graph. In this paper [Formula: see text] as the main theorem we prove that if [Formula: see text] is a prime power and satisfies some especial conditions [Formula: see text] then there exists a number [Formula: see text] associated to [Formula: see text] such that for all [Formula: see text] the finite linear simple group [Formula: see text] is quasirecognizable by prime graph. Finally [Formula: see text] by a calculation via a computer program [Formula: see text] we conclude that the above conjecture is valid for the simple group [Formula: see text] where [Formula: see text] [Formula: see text] is an odd number and [Formula: see text].

Symbolic computation study on exact solutions to a generalized (3+1)-dimensional Kadomtsev-Petviashvili-type equation

2020 ◽  
pp. 2150116
Author(s):  
Cheng-Cheng Zhou ◽  
Xing Lü ◽  
Hai-Tao Xu
Keyword(s):  
Numerical Simulation ◽  
Exact Solutions ◽  
Prime Number ◽  
Symbolic Computation ◽  
Three Dimensional ◽  
Type Equation ◽  
Two Dimensional ◽  
Type Solution ◽  
Bilinear Operators ◽  
Number Formula

Based on the prime number [Formula: see text], a generalized (3+1)-dimensional Kadomtsev-Petviashvili (KP)-type equation is proposed, where the bilinear operators are redefined through introducing some prime number. Computerized symbolic computation provides a powerful tool to solve the generalized (3+1)-dimensional KP-type equation, and some exact solutions are obtained including lump-type solution and interaction solution. With numerical simulation, three-dimensional plots, density plots, and two-dimensional curves are given for particular choices of the involved parameters in the solutions to show the evolutionary characteristics.

Indices in a number field II

2019 ◽  
Vol 15 (01) ◽  
pp. 89-103
Author(s):  
Mohamed Ayad ◽  
Rachid Bouchenna ◽  
Omar Kihel
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Additive Group ◽  
Necessary Condition ◽  
Ring Of Integers ◽  
Number Formula

Let [Formula: see text] be a number field of degree [Formula: see text] over [Formula: see text] and [Formula: see text] its ring of integers. For a prime number [Formula: see text], we determine the types of splittings of [Formula: see text] in [Formula: see text] for which the set [Formula: see text] is of cardinality a power of [Formula: see text]. We prove that this necessary condition is also sufficient for [Formula: see text] to be a subgroup of the additive group [Formula: see text]. Consequently, we show that, in this case, the subset of [Formula: see text], [Formula: see text] is an order of the number field.

Mixed cobinary trees

2018 ◽  
Vol 17 (09) ◽  
pp. 1850170
Author(s):  
Kiyoshi Igusa ◽  
Jonah Ostroff
Keyword(s):  
Short Proof ◽  
Point Of View ◽  
Catalan Number ◽  
Binary Trees ◽  
Cluster Theory ◽  
Type Formula ◽  
Isomorphism Classes ◽  
Basic Cluster ◽  
Number Formula ◽  
Special Case

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees (MCTs). We show that the number of isomorphism classes of such trees is given by the Catalan number [Formula: see text] where [Formula: see text] is the number of internal nodes. We also consider the corresponding quiver [Formula: see text] of type [Formula: see text]. As a special case of more general known results about the relation between [Formula: see text]-vectors, representations of quivers and their semi-invariants, we explain the bijection between MCTs and the vertices of the generalized associahedron corresponding to the quiver [Formula: see text]. These results are extended to [Formula: see text]-clusters in the next paper. We give one application: a new short proof of a conjecture of Reineke using MCTs.

scholarly journals Period and toroidal knot mosaics

2017 ◽  
Vol 26 (05) ◽  
pp. 1750031 ◽  
Author(s):  
Seungsang Oh ◽  
Kyungpyo Hong ◽  
Ho Lee ◽  
Hwa Jeong Lee ◽  
Mi Jeong Yeon
Keyword(s):  
Quantum System ◽  
Prime Number ◽  
Growth Rates ◽  
Exact Enumeration ◽  
Common Features ◽  
System A ◽  
Number Formula ◽  
And Mathematics ◽  
Definition Of ◽  
Positive Integers

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot [Formula: see text]-mosaic is an [Formula: see text] matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot [Formula: see text]-mosaics for any positive integers [Formula: see text] and [Formula: see text], toroidal knot [Formula: see text]-mosaics for co-prime integers [Formula: see text] and [Formula: see text], and furthermore toroidal knot [Formula: see text]-mosaics for a prime number [Formula: see text]. We also analyze the asymptotics of the growth rates of their cardinality.

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