Isomorphic Groups Research Articles

Зображення формальними матрицями елементiв матричних груп над асоцiативними кiльцями

The paper substantiates the image by formal matrices of some elements of linear groups over associative rings by using the properties of redundant and fixed modules. The form of images of elements of linear groups with respect to their homomorphisms in the group of automorphisms of modules over associative rings of 1, in which there are reversible elements 2 or 3, is shown. Using this approach, the authors described homomorphisms with the condition (*) of matrix groups over associative rings with 1. In particular, the isomorphisms of matrix groups over associative rings are homomorphisms with condition (*).

The twisted group ring isomorphism problem over fields

Similarly to how the classical group ring isomorphism problem asks, for a commutative ring $R$, which information about a finite group $G$ is encoded in the group ring $RG$, the twisted group ring isomorphism problem asks which information about $G$ is encoded in all the twisted group rings of $G$ over $R$. We investigate this problem over fields. We start with abelian groups and show how the results depend on the roots of unity in $R$. In order to deal with non-abelian groups we construct a generalization of a Schur cover which exists also when $R$ is not an algebraically closed field, but still linearizes all projective representations of a group. We then show that groups from the celebrated example of Everett Dade which have isomorphic group algebras over any field can be distinguished by their twisted group algebras over finite fields.

Pseudo-random scalar multiplication based on group isomorphism

Elliptic curve cryptography is an essential cryptography which is widely used in data encryption, key agreement, digital signature, and other applications. Scalar multiplication is the fundamental operation in all ECC schemes, such as ECDSA, ECIES, and ECMQV. Even a slight improvement of scalar multiplication is precious. Various methods have been proposed for improving efficiency of the scalar multiplication, including the windows method, the NAF method, and the w-NAF method. However, the endeavour in this direction almost exhausted in the past several years since it is hard to find a method substantially better than the w-NAF method. This paper focuses on the scalar multiplication algorithm for the case when the scalar is a pseudo-random number. A faster pseudo-random scalar multiplication method is proposed based on a group isomorphism between the pseudo-random number group and elliptic curve point group. Experimental results show that the proposed method can considerably reduce the computation time compared with those traditional methods. The pseudo-random scalar multiplication accounts for a significant proportion of total scalar multiplication operations in almost all ECC schemes. Therefore, the proposed method is promising and applicable for various ECC applications in the fields such as Internet of things, edge computing, and swarm robotics.

The fundamental solution to $$\Box _b$$ on quadric manifolds: part 2. $$L^p$$ regularity and invariant normal forms

This paper is the second of a multi-part series in which we explore geometric and analytic properties of the Kohn–Laplacian and its inverse on general quadric submanifolds of $${\mathbb{C}}^n\times {\mathbb{C}}^m$$ . We have two goals in this paper. The first is to give useable sufficient conditions for a map T between quadrics to be a Lie group isomorphism that preserves $$\Box _b$$ , and the second is to establish a framework for which appropriate derivatives of the complex Green operator are continuous in $$L^p$$ and $$L^p$$ -Sobolev spaces (and hence are hypoelliptic). We apply the general results to codimension two quadrics in $${\mathbb{C}}^4$$ .

Classification of Sylow classes of parabolic and reflection subgroups in unitary reflection groups

Let be a prime divisor of the order of a finite unitary reflection group. We classify up to conjugacy the parabolic and reflection subgroups that are minimal with respect to inclusion, subject to containing an -Sylow subgroup. The classification assists in describing the -Sylow subgroups of unitary reflection groups up to group isomorphism. This classification also relates to the modular representation theory of finite groups of Lie type. We observe that unless a parabolic subgroup minimally containing an -Sylow subgroup is G itself, the reflection subgroup within the parabolic minimally containing an -Sylow subgroup is the whole parabolic subgroup.

Polymer Dynamics Through Group Invariance of SL(2R) - Type in a Fractal Paradigm

We analyze polymer dynamics in a fractal paradigm. Then, it is shown that polymer dynamics in the form of Schrödinger - type regimes imply synchronization processes of the polymers� structural units, through joint invariant function of two simultaneous isomorphic groups of SL(2R) - type, as solutions of Stoka equations. In this context, period doubling, damped oscillations, self - modulation and chaotic regimes emerge as natural behaviors in the polymer dynamics. The present model can also be applied to a large class of materials, such as biomaterials, biocomposites and other advanced materials.

Hazewinkel Functional Lemma and Classification of Formal Groups

The main fields of application of formal groups are algebraic geometry and class field theory. The later uses both the classical Hilbert symbol (the norm-residue symbol) and its generalization. One of the most important problems is finding explicit formulas for various modifications of this symbol related to formal groups. There are two approaches to constructing formal groups (i.e., power series satisfying certain conditions). The functional lemma proved by Hazewinkel allows one to make formal groups with coefficients from a ring of characteristic zero by means of functional equations using a certain ideal of this ring, an overfield of the ring, and a ring homomorphism with specified properties (e.g., identical; for a local field, the Frobenius homomorphism can be chosen). There is a convenient criterion for the isomorphism of formal groups, constructed by Hazewinkel’s formula, as well as a formula for logarithms (in particular, the Artin–Hasse logarithm). At the same time, Lubin and Tate construct formal groups over local fields, using isogeny, and Honda, when constructing formal groups over the ring of integers of a discrete valued field of characteristic zero, introduces a certain noncommutative ring induced by the original ring and a fixed homomorphism. The paper relates the classical classification of formal groups (standard, generalized, relative Lubin–Tate formal groups, and formal Honda groups) to their classification using the Hazewinkel functional lemma. For each type, the corresponding functional equations are composed and logarithms, as well as series used to construct an explicit formula for the Hilbert symbol, are studied.

A hyperspace of convex bodies arising from tensor norms

In a preceding work it is determined when a convex body in Rd, d=d1⋯dl, is the unit ball of a reasonable crossnorm on Rd1⊗⋯⊗Rdl. Consequently, the class of tensorial bodies is introduced and a Banach-Mazur type compactum is proved to exist. In this paper, we introduce the hyperspace of these convex bodies. We called “the space of tensorial bodies”. It is proved that the group of linear isomorphisms on Rd1⊗⋯⊗Rdl preserving decomposable vectors acts properly on it. A compact global slice for the space is constructed. Then, topological representatives for the space of tensorial bodies and the Banach-Mazur type compactum are given. It is showed that the set of ellipsoids in the class of tensorial bodies is homeomorphic to the Euclidean space of dimension p=d1(d1+1)2+⋯+dl(dl+1)2. We also prove the continuity of both the projective and the injective tensor product of 0-symmetric convex bodies, with respect to the Hausdorff distance.

Towards Stochasticity through Joint Invariant Functions of Two Isomorphic Lie Algebras of SL(2R) Type

In the motion fractal theory, the scale relativity dynamics of any complex system are described through various Schrödinger or hydrodynamic type fractal “regimes”. In the one dimensional stationary case of Schrödinger type fractal “regimes”, synchronizations of complex system entities implies a joint invariant function with the simultaneous action of two isomorphic groups of the S L ( 2 R ) type as solutions of Stoka type equations. Among these joint invariant functions, Gaussians become in the Jeans’s sense, probability density (i.e., stochasticity) whenever the information on the complex system analyzed is fragmentary. In the two-dimensional case of hydrodynamic type fractal “regimes” at a non-differentiable scale, the soliton and soliton-kink of fractal type of the velocity field generate the minimal vortex of fractal type that becomes the source of all turbulences in the complex systems dynamics. Some correlations of our model to experimental data were also achieved.

To the Question of the Root-Class Residuality of Free Constructions of Groups

Let $$\mathcal{C}$$ be a root class of groups and $$\mathcal{\pi}_{1}(\mathcal{G})$$ be the fundamental group of a graph $$\mathcal{G}$$ of groups. We prove that if $$\mathcal{G}$$ has a finite number of edges and there exists a homomorphism of $$\mathcal{\pi}_{1}(\mathcal{G})$$ onto a group of $$\mathcal{C}$$ acting injectively on all the edge subgroups, then $$\mathcal{\pi}_{1}(\mathcal{G})$$ is residually a $$\mathcal{C}$$ -group. The main result of the paper is that the inverse statement is not true for many root classes of groups. The proof of this result is based on the criterion for the fundamental group of a graph of isomorphic groups to be residually a $$\mathcal{C}$$ -group, which is of independent interest.

A Systematic Approach to Group Properties Using its Geometric Structure

The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic groups to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph’s vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.

A Systematic Approach to Group Properties Using its Geometric Structure

The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic groups to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph’s vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.

Complex group algebras of almost simple unitary groups

The aim of this article is to contribute to a question of R. Brauer that “when do non-isomorphic groups have isomorphic complex group algebras?” Let H and G be finite groups where and let denote the first column of the complex character table of H. In this article, we show that if then provided that q + 1 divides neither n nor n – 1. Consequently, it is shown that G is uniquely determined by the structure of its complex group algebra. This in particular extends a recent result of Bessenrodt et al. [Algebra Number Theory 9 (2015), 601–628] to the almost simple groups of arbitrary rank.

FINITE DOMAIN STRUCTURES IN THE FRAMEWORK OF THE CONCEPT OF A MODEL-THEORETIC PROPERTY

Abstract. In this work, we follow the algebraic approach using definability by formulas presentable in both existential and universal forms. The class of algebraic Cartesian interpretations of theories is studied presenting a foundation of the finitary first-order combinatorics. Common properties of first-order definability in finite models are studied. Some relations are obtained between automorphism groups of finite models and isomorphisms of Cartesian extensions of their theories. A formal definition of the notion of a model-theoretic property is analyzed based on a separate consideration of cases of theories with finite and infinite models. A description of model-theoretic properties defined via finite domains is found. It is established that the class of all finite models with first-order definable elements as well as the corresponding class of theories of such models forms the only model-theoretic property and, therefore, is of little interest as a database with an interface based on the first-order logic language. Key words: first-order logic, Cartesian extension of a theory, Tarski-Lindenbaum algebra, model-theoreticproperty, computable isomorphism.

On special subgroups of fundamental group

Suppose ? is a nonzero cardinal number, I is an ideal on arc connected Topological space X, and B?I(X) is the subgroup of ?1(X) (the first fundamental group of X) generated by homotopy classes of ?_I loops. The main aim of this text is to study B?I(X)s and compare them. Most interest is in ? ? {?,c} and I ? {Pfin(X), {?}}, where Pfin(X) denotes the collection of all finite subsets of X. We denote B?{?}(X) with B?(X). We prove the following statements: for arc connected topological spaces X and Y if B?(X) is isomorphic to B?(Y) for all infinite cardinal number ?, then ?1(X) is isomorphic to ?1(Y); there are arc connected topological spaces X and Y such that ?1(X) is isomorphic to ?1(Y) but B?(X) is not isomorphic to B?(Y); for arc connected topological space X we have B?(X) ? Bc(X) ? ?1(X); for Hawaiian earring X, the sets B?(X), Bc(X), and ?1(X) are pairwise distinct. So B?(X)s and B?I(X)s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.

Algebraic Characteristics of Anti-Intuitionistic Fuzzy Subgroups Over a Certain Averaging Operator

In this paper, we propose the concept of ρ anti-intuitionistic fuzzy sets, ρ anti - intuitionistic fuzzy subgroups and prove some of their algebraic properties. We investigate a necessary and sufficient condition for a ρ-anti intuitionistic fuzzy set to be a ρ-anti intuitionistic fuzzy subgroup. We extend this ideology by defining the notions of ρ anti-intuitionistic fuzzy coset, ρ anti-intuitionistic fuzzy normal subgroup and derive some of their key algebraic characteristics. In addition, we study the quotient group of a group induced by ρ-anti intuitionistic fuzzy normal subgroup and establish a group isomorphism between this newly defined quotient group and the quotient group of group G relative to its particular normal subgroup G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Sρ</sub> .

Analisis Kemampuan Mahasiswa dalam Menyelesaikan Soal Pembuktian tentang Isomorfisma Grup

Group Isomorphism is one of the sub topic in the Algebra Structure. This sub topic required prerequisite material about the bijective function. If this prerequisite material is not mastered, it will be difficult to study Isomorphism Group material. There are still many students who found difficulties to solve probative questions about Group Isomorphism. The major problem are they were forgetting and not understanding the prerequisite material. Therefore, it should be a further research on the ability of students to prove the questions of proof and errors in preparing evidence about Group Isomorphism.

K-theory of affine actions

For a Lie group $G$ and a vector bundle $E$ we study those actions of the Lie group $TG$ on $E$ for which the action map $TG\times E \to E$ is a morphism of vector bundles, and call those \emph{affine actions}. We prove that the category $\mathrm{Vect}_{TG}^{\mathrm{aff}}\left(X\right)$ of such actions over a fixed $G$-manifold $X$ is equivalent to a certain slice category $\mathfrak g_X \backslash \mathrm{Vect}_G\left(X\right)$. We show that there is a monadic adjunction relating $\mathrm{Vect}_{TG}^{\mathrm{aff}}\left(X\right)$ to $\mathrm{Vect}_G\left(X\right)$, and the right adjoint of this adjunction induces an isomorphism of Grothendieck groups $K_{TG}^{\mathrm{aff}}\left(X\right) \cong KO_G\left(X\right)$. Complexification produces analogous results involving $T_\mathbb C G$ and $K_G\left(X\right)$.

Light groups of isomorphisms of Banach spaces and invariant LUR renormings

Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.

New Exact and Heuristic Algorithms for Graph Automorphism Group and Graph Isomorphism

We describe five new algorithms, named Vsep. Four of them are for the graph automorphism group and the fifth one is for finding an isomorphism between two graphs. All nonequivalent terminal nodes-discrete partitions of the search tree are stored. This is the main difference of the exact version with the known algorithms for graph automorphism group. A new strategy is used in the exact algorithm: if during its execution the computed stabilizer orbits and order get wrong values, then the algorithm continues from a new starting point losing some of the results determined so far. The new starting point is such that the results are correct. The proposed algorithms have been tested on well-known benchmark graphs and compared with three of the most competitive known tools. The results show that for some graph families the exact Vsep algorithm outperforms these algorithms, and vice-versa for some of the others. None of the tested algorithms outperform others in all cases. The heuristic versions use reduced search trees. They are almost exact and are faster than the exact one with very rare exceptions. They are applied mainly for regular graphs. The heuristic algorithms are a new choice for the user. The experiments show that the running times of Vsep algorithms have a slight dependence on vertex labeling.