Isomorphic Groups Research Articles

The commutative inverse semigroup of partial abelian extensions

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an -partial Galois extension of and a normal subgroup H of G, we prove that induces a unital partial action of G/H on the subalgebra of invariants of S such that is an -partial Galois extension of Second, assuming that G is abelian, we construct a commutative inverse semigroup whose elements are equivalence classes of -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between and T(G, A), where ρ is a congruence on and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of reduces to the case where G is cyclic. The set of idempotents of is also investigated.

Arbitrarily large [formula omitted]-Morita Frobenius numbers

We construct blocks of finite groups with arbitrarily large O-Morita Frobenius numbers. There are no known examples of two blocks defined over O, with isomorphic defect groups, that are not Morita equivalent but the corresponding blocks defined over k are. Therefore, the above strongly suggests that Morita Frobenius numbers are also unbounded, which would answer a question of Benson and Kessar.

Homotopy type of planar Peano continua

We prove that the homotopy equivalence class of a map from a Peano continuum into a planar or one-dimensional Peano continuum is determined by the induced homomorphism of fundamental groups. This provides a new proof that planar sets are aspherical and is used to prove that two planar or one-dimensional Peano continua are homotopy equivalent if and only if they have isomorphic fundamental groups.

On the Exponent of the Weak Commutativity Group $$\chi (G)$$

The weak commutativity group $\chi(G)$ is generated by two isomorphic groups $G$ and $G^{\varphi }$ subject to the relations $[g,g^{\varphi}]=1$ for all $g \in G$. The group $\chi(G)$ is an extension of $D(G) = [G,G^{\varphi}]$ by $G \times G$. We prove that if $G$ is a finite solvable group of derived length $d$, then $\exp(D(G))$ divides $\exp(G)^{d}$ if $|G|$ is odd and $\exp(D(G))$ divides $2^{d-1}\cdot \exp(G)^{d}$ if $|G|$ is even. Further, if $p$ is a prime and $G$ is a $p$-group of class $p-1$, then $\exp(D(G))$ divides $\exp(G)$. Moreover, if $G$ is a finite $p$-group of class $c\geq 2$, then $\exp(D(G))$ divides $\exp(G)^{\lceil \log_{p-1}(c+1)\rceil}$ ($p\geq 3$) and $\exp(D(G))$ divides $2^{\lfloor \log_2(c)\rfloor} \cdot \exp(G)^{\lfloor \log_2(c)\rfloor+1}$ ($p=2$).

Coarse groups, and the isomorphism problem for oligomorphic groups

Let [Formula: see text] denote the topological group of permutations of the natural numbers. A closed subgroup G of [Formula: see text] is called oligomorphic if for each n, its natural action on n-tuples of natural numbers has only finitely many orbits. We study the complexity of the topological isomorphism relation on the oligomorphic subgroups of [Formula: see text] in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup [Formula: see text] of [Formula: see text], the coarse group [Formula: see text] is the structure with domain the cosets of open subgroups of G, and a ternary relation [Formula: see text]. This structure derived from G was introduced in [A. Kechris, A. Nies and K. Tent, The complexity of topological group isomorphism, J. Symbolic Logic 83(3) (2018) 1190–1203, Sec. 3.3 ]. If G has only countably many open subgroups, then [Formula: see text] is a countable structure. Coarse groups form our main tool in studying such closed subgroups of [Formula: see text]. We axiomatize them abstractly as structures with a ternary relation. For the oligomorphic groups, and also the profinite groups, we set up a Stone-type duality between the groups and the corresponding coarse groups. In particular, we can recover an isomorphic copy of G from its coarse group in a Borel fashion. We use this duality to show that the isomorphism relation for oligomorphic subgroups of [Formula: see text] is Borel reducible to a Borel equivalence relation with all classes countable. We show that the same upper bound applies to the larger class of closed subgroups of [Formula: see text] that are topologically isomorphic to oligomorphic groups.

On the Group of Holomorphic Automorphisms of a Model Surface

In this paper, it is proved that the group $$\mathrm{Aut}\, Q$$ of all holomorphic automorphisms of a holomorphically homogeneous nondegenerate model surface $$Q$$ is a subgroup of the group of birational isomorphisms of the ambient complex space (the Cremona group) of uniformly bounded degree. The degree is estimated in terms of the dimension of the ambient space (Theorem 4). It is shown that no condition of the theorem can be weakened. In the paper, the question of the connectivity of $$\mathrm{Aut}\, Q$$ is also considered (Theorem 7). This paper is directly adjacent to the previous paper of the author [7].

A Bakry–Émery Almost Splitting Result With Applications to the Topology of Black Holes

The almost splitting theorem of Cheeger-Colding is established in the setting of almost nonnegative generalized $m$-Bakry-\'{E}mery Ricci curvature, in which $m$ is positive and the associated vector field is not necessarily required to be the gradient of a function. In this context it is shown that with a diameter upper bound and volume lower bound the fundamental group of such manifolds is almost abelian. Furthermore, extensions of well-known results concerning Ricci curvature lower bounds are given for generalized $m$-Bakry-\'{E}mery Ricci curvature. These include: the first Betti number bound of Gromov and Gallot, Anderson's finiteness of fundamental group isomorphism types, volume comparison, the Abresch-Gromoll inequality, and a Cheng-Yau gradient estimate. Finally, this analysis is applied to stationary vacuum black holes in higher dimensions to find that low temperature horizons must have limited topology, similar to the restrictions exhibited by (extreme) horizons of zero temperature.

As similaridades estruturais decorrentes dos efeitos do isomorfismo mimético nos programas de Pós-Graduação em Administração

Este estudo teve como objetivo identificar a presença de isomorfismo mimético a partir das semelhanças estruturais dos currículos dos Programas de Pós-Graduação em Administração-PPGA, e se o mimetismo contribui para um desempenho semelhante no ranking da CAPES. Metodologicamente, utilizou-se estatística descritiva e análise de agrupamento com a técnica de Co-plot como procedimento de apresentação e análise de dados. Como resultado, identificamos quatro grupos de PPGAs e a similaridade estrutural entre eles. Percebemos também que a formação de clusters, a identificação de um alto nível de similaridade entre os programas e a aceitação parcial de pertencer a determinado grupo podem se refletir no desempenho do PPGP. Conclui-se que a postura diferenciada de cada grupo observado de cada grupo isomórfico não significa desempenho semelhante a todos os componentes. Portanto, argumentamos que não é evidente obter os mesmos resultados do espelhamento de estruturas e procedimentos das organizações de referência.

Conceptual metaphors for isomorphism and homomorphism: Instructors’ descriptions for themselves and when teaching

Group isomorphism and homomorphism are topics central to abstract algebra, yet research on instructors’ views of these concepts is limited. Based on interviews from two instructors as well as classroom video from eight class periods, this paper examines the language used to discuss isomorphism and homomorphism. Language used by instructors in interviews and classroom settings are identified and classified into four main categories: formal definition, mapping, sameness, and combinations of sameness and mapping language. How the two instructors drew on language classified into those four categories in the interview and instruction settings are examined for isomorphism and homomorphism. Similarities and differences between the interview and instruction contexts reveal the wide variety of ways of understanding isomorphism and homomorphism as well as a research need to examine mathematicians’ content knowledge in more than one context.

Group presentations for links in thickened surfaces

Using a combinatorial argument, we prove the well-known result that the Wirtinger and Dehn presentations of a link in 3-space describe isomorphic groups. The result is not true for links [Formula: see text] in a thickened surface [Formula: see text]. Their precise relationship, as given in [R. E. Byrd, On the geometry of virtual knots, M.S. Thesis, Boise State University (2012)], is established here by an elementary argument. When a diagram in [Formula: see text] for [Formula: see text] can be checkerboard shaded, the Dehn presentation leads naturally to an abelian “Dehn coloring group,” an isotopy invariant of [Formula: see text]. Introducing homological information from [Formula: see text] produces a stronger invariant, [Formula: see text], a module over the group ring of [Formula: see text]. The authors previously defined the Laplacian modules [Formula: see text] and polynomials [Formula: see text] associated to a Tait graph [Formula: see text] and its dual [Formula: see text], and showed that the pairs [Formula: see text], [Formula: see text] are isotopy invariants of [Formula: see text]. The relationship between [Formula: see text] and the Laplacian modules is described and used to prove that [Formula: see text] and [Formula: see text] are equal when [Formula: see text] is a torus.

On the existence of specialization isomorphisms of admissible fundamental groups in positive characteristic

On the existence of specialization isomorphisms of admissible fundamental groups in positive characteristic

Geometric Variational Finite Element Discretizations for Fluids

Abstract We present an overview of finite element variational integrators for compressible and incompressible fluids with variable density. The numerical schemes are derived by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. In this setting, discrete vector fields correspond to a certain subspace of the Lie algebra of this group. This subspace is shown to be isomorphic to a Raviart-Thomas finite element space. We illustrate the conservation properties of the scheme with the Rayleigh-Taylor instability test.

ОБ АЛГОРИТМИЧЕСКОЙ РАЗРЕШИМОСТИ ПРОБЛЕМЫ ОБОБЩЕННОЙ СОПРЯЖЕННОСТИ ПОДГРУПП В ГРУППАХ КОКСТЕРА С ДРЕВЕСНОЙ СТРУКТУРОЙ

The main algorithmic problems of combinatorial group theory posed by M. Den and G. Titze at the beginning of the twentieth century are the problems of word, word conjugacy and of group isomorphism. However, these problems, as follows from the results of P.S. Novikov and S.I. Adyan, turned out to be unsolvable in the class of finitely defined groups. Therefore, algorithmic problems began to be considered in specific classes of groups. The word conjugacy problem allows for two generalizations. On the one hand, we consider the problem of conjugacy of subgroups, that is, the problem of constructing an algorithm that allows for any two finitely generated subgroups to determine whether they are conjugate or not. On the other hand, the problem of generalized conjugacy of words is posed, that is, the problem of constructing an algorithm that allows for any two finite sets of words to determine whether they are conjugated or not. Combining both of these generalizations into one, we obtain the problem of generalized conjugacy of subgroups. Coxeter groups were introduced in the 30s of the last century, and the problems of equality and conjugacy of words are algorithmically solvable in them. To solve other algorithmic problems, various subclasses are distinguished. This is partly due to the unsolvability in Coxeter groups of another important problem – the problem of occurrence, that is, the problem of the existence of an algorithm that allows for any word and any finitely generated subgroup of a certain group to determine whether this word belongs to this subgroup or not. The paper proves the algorithmic solvability of the problem of generalized conjugacy of subgroups in Coxeter groups with a tree structure.

ХРОМАТИЧНЕ ЧИСЛО ФУНКЦІЇ

Побудова на досліджуваній множині математичних об'єктів групової структури і використання її властивостей є одним з ефективних методів дослідження. Одним з центральних понять теорії груп є поняття гомоморфізму, яке виявляється дуже корисним при вивченні властивостей груп. Гомоморфізм - це відображення з однієї групи в іншу, яке зберігає групову операцію. У даній статті авторами побудований аналог поняття гомоморфізму на випадок, коли замість групової операції задається довільне, усюди визначене відображення f X X n : → . У статті докладно розглядається випадок, коли n = 2 і X ⊂ R . Дається визначення хроматичного числа цього відображення і наводяться приклади його обчислення. Наведені приклади хроматичних чисел деяких груп з необхідними поясненнями. Введено поняття хроматичного числа дійсної числової функції і показано, що це поняття тісно пов'язане з поняттям R - функції В.Л. Рвачева. Спираючись на відомі раніше результати, показано, що існують числові функції з нескінченними хроматичними числами. Для прикладу наведено хроматичні числа деяких функцій, дані пояснення отриманих результатів. Основним результатом цієї статті є доведення того факту, що лінійна функція двох дійсних змінних f (x, y) =αx + βy +γ ,αβ ≠ 0 не має скінченого хроматичного числа. Аналогічний результат доведений для функції ( ) 2 2 g x, y = x − y двох дійсних змінних. Таким чином, множину R неможна розфарбувати в скінчене число кольорів так, щоб колір значення функції αx + βy +γ , де αβ ≠ 0 однозначно визначався кольором її аргументів. Те ж стосується функцій 2 2 x − y і ax x x b n ... + 1 2 , де n >1, ab ≠ 0 . У термінах R - функцій отриманий результат можна сформулювати наступним чином: функції f (x, y) і g(x, y) (як і функція ax x x b n ... + 1 2 при n >1, ab ≠ 0 ) не можуть бути R - функціями ні при якому виборі супроводжуючих функцій багатозначної логіки. Таким чином, в даній статті введено поняття хроматичного класу і хроматичного числа функції. Знайдено зв'язок між отриманими поняттями і теорією груп. Продемонстровано, що поняття хроматичного числа функції на деякій множині тісно пов'язане з поняттям R - функції В.Л. Рвачева. Відзначено, що для доведення неізоморфності груп можна використовувати той факт, що для ізоморфних груп хроматичні числа і хроматичні класи, до яких вони належать, збігаються.

Finitely Generated Nilpotent Groups of Infinite Cyclic Commutator Subgroups

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial ℤ-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial ℤ-groups, we finally obtain the structure and invariants of the group G.

Гомоморфiзми з умовою (*), якщо 2 – оборотний елемент

Вивчення гомоморфiзмiв матричних груп над асоцiативними кiльцями розпочалося майже 100 рокiв тому роботами Шраєра i Ван-дер-Вардена i в подальшому розвивалися в працях Дьєдоне, Хуа Ло-гена, Райнера, О’Мiри, Хана, Ю.I. Мерзлякова, Уотерхауса, О.В. Мiхальова, Ю.I. Зельманова, I.З. Голубчика, В.М. Петечука та iнших авторiв. В основi вивчення знаходяться груповi властивостi повної лiнiйної групи GL(n, R) – множини всiх оборотних матриць над асоцiативним кiльцем R з 1. При n ≥ 3 у всiх вiдомих випадках, незважаючи на вiдмiннiсть методiв, якi застосовувалися, автоморфiзми повної лiнiйної групи виявлялись добутком стандартних автоморфiзмiв. Саме оборотнiсть елемента 2 давала можливiсть розглядати все бiльш широкi класи кiлець над якими можливий стандартний опис гомоморфiзмiв матричних груп. Якщо 2 – необоротний елемент, то при n ≥ 3 В.М. Петечук зробив опис автоморфiзмiв групи GL(n, R) у випадку, коли R – комутативне локальне кiльце. Виявилося, що при n ≥ 4 всi автоморфiзми таких груп є добутком стандартних автоморфiзмiв, а при n = 3 їх можна виразити через стандартнi i деякий нестандартний автоморфiзми. Спираючись на цей результат, В.М. Петечук [2] отримав опис iзоморфiзмiв групи GL(n, R), n ≥ 3, якщо R – довiльне комутативне кiльце. Зокрема, вiн здiйснив опис гомоморфiзмiв Λ : P E (n, R) → P GL(m, K), m ≥ 3, n ≥ 3 таких, що ΛP E (n, R) = P H i H ⊇ E (m, K) над довiльними комутативними кiльцями R i K. I.З. Голубчик i О.В. Мiхальов [3], використовуючи системи iдемпотентiв, i незалежно Ю.I. Зельманов [4], використовуючи методи йорданових алгебр, отримали опис iзоморфiзмiв групи E (n, R), n ≥ 3, 2 ∈ R∗ на групу E (m, K), 2 ∈ K∗ над довiльними асоцiативними кiльцями R i K з 1. В.М. Петечук [5] зробив опис гомоморфiзмiв групи P E (n, R), n ≥ 3 в групу GL(m, K), m ≥ 2, 2 ∈ K∗ у випадку, коли нерухомi пiдмодулi деяких елементiв четвертого порядку збiгаються з нерухомими пiдмодулями їх квадратiв. З нього випливають результати I.З. Голубчика, О.В. Мiхальова i Ю.I. Зельманова. Розвиваючи технiку, пов’язану з iдемпотентами, I.З. Голубчик [6] здiйснив опис iзоморфiзмiв груп GL(n, R) i GL(m, K) при n, m ≥ 4 над асоцiативними кiльцями R i K. Виявилося, що вони допускають стандартний опис на групi E (n, R). Авторами В.М. Петечук, Ю.В. Петечук [7, 8] описанi гомоморфiзми з умовою (*) з чого зокрема випливає i опис iзоморфiзмiв повних лiнiйних груп над асоцiативними кiльцями. У данiй роботi удосконалюються i розширюються методи опису гомоморфiзмiв з умовою (*), якщо елемент 2 є оборотним в кiльцi K i n ≥ 3. Основним результатом роботи є наступна теорема. Нехай R i K – асоцiативнi кiльця з 1, 2 ∈ K∗ , E (n, R) ⊆ G ⊆ GL(n, R), n ≥ 3, W – лiвий K-модуль, гомоморфiзм Λ : G → GL(W) задовольняє умову (*). Тодi Λ має стандартний опис на групi E (n, R).

The number of affine equivalent classes and extended affine equivalent classes of vectorial Boolean functions

Affine equivalent classes and extended affine equivalent (EA-equivalent for short) classes of vectorial Boolean functions have important applications in cryptography, logic circuit, sequences for communications, etc. Recently, Y. Zhang et al., computed the number of affine equivalent classes of n-variable Boolean functions when 1≤n≤10 by group isomorphism (Zhang et al., 2016). However, the case for affine equivalent vectorial Boolean function remains a challenging open problem. Furthermore, little result for EA-equivalent vectorial Boolean function is known except a trivial lower bound mentioned in Mullen and Panario (2013, P246). In this paper, we focus on the challenging problem of calculating the number of affine equivalent classes and EA-equivalent classes of vectorial Boolean functions. First, for EA-equivalence, we prove that the trivial lower bound proposed in Mullen and Panario (2013) has at least 3 effective figures if n is not too small. We then show that the lower bound also holds for affine equivalent classes. Furthermore, we give an explicit formula and calculate the exact number of affine equivalent classes of (n,m)-functions with 1≤m,n≤11 by GAP. The results in this paper are helpful for the theory and applications of the classifications of vectorial Boolean functions.

A Variational Finite Element Discretization of Compressible Flow

We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure-preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. In this setting, discrete vector fields correspond to a certain subspace of the Lie algebra of this group. This subspace is shown to be isomorphic to a Raviart–Thomas finite element space. The resulting finite element discretization corresponds to a weak form of the compressible fluid equation that does not seem to have been used in the finite element literature. It extends previous work done on incompressible flows and at the lowest order on compressible flows. We illustrate the conservation properties of the scheme with some numerical simulations.

On categorical equivalence of finite p-rings

We prove that finite categorically equivalent p-rings have isomorphic additive groups (in particular, they have the same cardinality) and that the number of generators is a categorical invariant for finite rings. We also classify rings of size p 3 up to categorical equivalence.

Recovering algebraic curves from L-functions of Hilbert class fields

In this paper, we prove that a smooth hyperbolic projective curve over a finite field can be recovered from L-functions associated to the Hilbert class field of the curve and its constant field extensions. As a consequence, we give a new proof of a result of Mochizuki and Tamagawa that two such curves with isomorphic fundamental groups are themselves isomorphic.