Quotient Group Research Articles

Relative Commutativity Degree of Nonabelian Metabelian Groups of Order 32

A metabelian group is a group whose commutator subgroup is abelian. Similarly, a group G is metabelian if and only if there exists an abelian normal subgroup, A, such that the quotient group, G / A , is abelian. The scope of this research is only for nonabelian metabelian groups of order 32. The commutativity degree of a group G is the probability that two elements of the group G (chosen randomly with replacement) commute. This probability can be used to measure how close a group is to be abelian. This concept has been extended to the co-prime probability which is defined as the probability of a random pair of elements x and y in G for which the greatest common divisor for the order of x and order of y is equal to one. Furthermore, the study of relative commutativity degree of a subgroup H of a group G which is the probability of an element in H commutes with an element in G is included in this research. Previous researchers have determined the commutativity degree of nonabelian metabelian groups of order at most 32. Meanwhile, the co-prime probability and the relative commutativity degree of both cyclic and noncyclic subgroups H are obtained for nonabelian metabelian groups of order at most 30. Since there is no nonabelian group of order 31, thus in this research the co-prime probability and the relative commutativity degree of cyclic subgroups for nonabelian metabelian groups of order 32 are determined.

First Betti Numbers of the Orbits of Morse Functions on Surfaces

Let M be a connected compact orientable surface and let P be either the real line ℝ or a circle S1. The group D(M) of diffeomorphisms on M acts in the space of smooth mappings C∞(M,P) by the rule (f, h) ⟼ f ° h, where h ∈ D(M) and f ∈ C∞(M,P). For f ∈ C∞(M,P), let O(f) denote the orbit of f relative to the specified action. By ℳ(M,P) we denote the set of isomorphism classes for the fundamental groups 𝜋1O(f) of orbits of all Morse mappings f : M → P. S. Maksymenko and B. Feshchenko studied the sets of isomorphism classes ℬ and 𝒯 of groups generated by direct products and certain wreath products. They proved the inclusions ℳ(M,P) ∁ B valid under the condition that M differs from the 2-sphere S2 and 2-torus T2 and ℳ(T2,ℝ) ∁ T . We show that these inclusions are equalities and describe some subclasses of M(M,P) under certain restrictions imposed on the behavior of functions on the boundary 𝜕M. We also prove that, for any group G ∈ ℬ (G ∈ T ), the center Z(G) and the quotient group with respect to the commutator subgroup G/[G,G] are free Abelian groups of the same rank, which can be easily found by using the geometric properties of a Morse mapping f such that 𝜋1O(f) ≃ G. In particular, this rank is the first Betti number of the orbit O(f) of the mapping f.

Patterson–Sullivan Measures and Growth of Relatively Hyperbolic Groups

We prove that for a relatively hyperbolic group, there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of the group. Under natural assumptions, a similar result holds for the critical exponent of a cusp-uniform action of the group on a hyperbolic metric space. As a corollary, we obtain that the critical exponent of a torsion-free geometrically finite Kleinian group can be arbitrarily approximated by those of proper quotient groups. This resolves a question of Dal’bo–Peigné–Picaud–Sambusetti. Our approach is based on the study of Patterson–Sullivan measures on Bowditch boundary of a relatively hyperbolic group and gives a series of results on growth functions of balls and cones.

Review on some constructions of cyclically ordered groups

We review some techniques of construction of new cyclically ordered groups from the old ones. Especially, we focus on the conditions of quotient group to be a cyclically ordered group.

Topological Decompositions of the Pauli Group and their Influence on Dynamical Systems

In the present paper we show that it is possible to obtain the well known Pauli group P = 〈X,Y,Z | X2 = Y2 = Z2 = 1,(YZ)4 = (ZX)4 = (XY )4 = 1〉 of order 16 as an appropriate quotient group of two distinct spaces of orbits of the three dimensional sphere S3. The first of these spaces of orbits is realized via an action of the quaternion group Q8 on S3; the second one via an action of the cyclic group of order four mathbb {Z}(4) on S3. We deduce a result of decomposition of P of topological nature and then we find, in connection with the theory of pseudo-fermions, a possible physical interpretation of this decomposition.

LOCAL RESOURCE-BASED POTENCIES IN PAMEKASAN REGENCY WITH ONE VILLAGE ONE PRODUCT APPROACH

This article describes the potential of the Pamekasan Regency area by promoting local-based resources (endogenous development). The method applied: literature search, calculation of Location Quotient, observation and Focus group discussion. The results showed, the potential of the northern region: cattle, shallots, fish processing, tobacco, sonok beef cultural tourism destinations, biopharmaca, cayenne pepper, vegetables, coconut, furniture, bamboo, siwalan, buju’ bersila bersiti tourism, catfish, tembang fish, shrimp paste. Southern region: salt, batik, tobacco, rice, anchovy, vanname shrimp, laying hens, processed fish (anchovies, selar) cattle fattening, tourism (jumiang beach, talang siring, fire never goes out), red chilies, krepek tette, marine tourism, oysters, mangrove, crab, small crab, seaweed, lorjuk, banana, rengginang, earthenware, sate lala ', krupuk (tack, fish, pulley), handicraft (bamboo, bag, skin), ampar stone religious tourism, broilers, tofu factories, peanuts, domestic chicken, herbal chilies, oranges, fresh fish restaurants, corn, shrimp, mackerel, herbal medicine, ball stadium, processed corn, green beans, drinks, vegetables, potoh, water tourism. Central region: agro-tourism (durian fruit, avocado klengkeng, mango, biopharmaca, Puncak Ratu Tour, Bukit Brukoh), batik, pebbles, cigarettes, pesantren-based economy, peanuts, crackers, plait, long beans, genting, bentoel, pepper, cayenne pepper, leeks, chips (cassava, taro, tette), beef, watermelon, cassava, coconut, banana, corn, brown sugar, chicken (laying eggs, domestic), siwalan, tape, rice, catfish.

HARISH-CHANDRA BIMODULES OVER QUANTIZED SYMPLECTIC SINGULARITIES

In this paper we classify the irreducible Harish-Chandra bimodules with full support over filtered quantizations of conical symplectic singularities under the condition that none of the slices to codimension 2 symplectic leaves has type E8. More precisely, consider the quantization 𝒜⋋ with parameter ⋋. We show that the top quotient \( \overline{\mathrm{HC}}\left(\mathcal{A}\lambda \right) \) of the category of Harish-Chandra 𝒜⋋-bimodules embeds into the category of representations of the algebraic fundamental group, Γ, of the open leaf. The image coincides with the representations of Γ/Γ⋋, where Γ⋋ is a normal subgroup of Γ that can be recovered from the quantization parameter ⋋ combinatorially. As an application of our results, we describe the Lusztig quotient group in terms of the geometry of the normalization of the orbit closure in almost all cases.

О двух свойствах группы Шункова

One of the interesting classes of mixed groups ( i.e. groups that can contain both elements of finite order and elements of infinite order) is the class of Shunkov groups. The group $G$ is called Shunkov group if for any finite subgroup $H$ of $G$ in the quotient group $N_G(H)/H$, any two conjugate elements of prime order generate a finite group. When studying the Shunkov group $G$, a situation often arises when it is necessary to move to the quotient group of the group $G$ by some of its normal subgroup $N$. In which cases is the resulting quotient group $G/N$ again a Shunkov group? The paper gives a positive answer to this question, provided that the normal subgroup $N$ is locally finite and the orders of elements of the subgroup $N$ are mutually simple with the orders of elements of the quotient group $G/N$. Let $ \mathfrak{X}$ be a set of groups. A group $G$ is saturated with groups from the set $ \mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $ G$ that is isomorphic to some group of $\mathfrak{X}$ . If all elements of finite orders from the group $G$ are contained in a periodic subgroup of the group $G$, then it is called the periodic part of the group $G$ and is denoted by $T(G)$. It is proved that the Shunkov group saturated with finite linear and unitary groups of degree 3 over finite fields has a periodic part that is isomorphic to either a linear or unitary group of degree 3 on a suitable locally finite field. \end{abstracte}

О периодических группах, насыщенных конечными группами Фробениуса

A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

О периодических группах Шункова с почти слойно конечными нормализаторами конечных подгрупп

Layer-finite groups first appeared in the work by S. N. Chernikov (1945). Almost layer-finite groups are extensions of layer-finite groups by finite groups. The class of almost layer-finite groups is wider than the class of layer-finite groups; it includes all Chernikov groups, while it is easy to give examples of Chernikov groups that are not layer-finite. The author develops the direction of characterizing well-known and well-studied classes of groups in other classes of groups with some additional (rather weak) finiteness conditions. A Shunkov group is a group 𝐺 in which for any of its finite subgroups 𝐾 in the quotient group <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>N</mi><mi>G</mi></msub><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mi>K</mi></mfrac></math> any two conjugate elements of prime order generate a finite subgroup. In this paper, we prove the properties of periodic not almost layer-finite Shunkov groups with condition: the normalizer of any finite nontrivial subgroup is almost layer-finite. Earlier, these properties were proved in various articles of the author, as necessary, sometimes under some conditions, then under others (the minimality conditions for not almost layer-finite subgroups, the absence of second-order elements in the group, the presence of subgroups with certain properties in the group). At the same time, it was necessary to make remarks that this property is proved in almost the same way as in the previous work, but under different conditions. This eliminates the shortcomings in the proofs of many articles by the author, in which these properties are used without proof.

On the fuzzification of Lagrange's theorem in $ (\alpha, \beta) $-Pythagorean fuzzy environment

<abstract> An $ (\alpha, \beta) $-Pythagorean fuzzy environment is an efficient tool for handling vagueness. In this paper, the notion of relative subgroup of a group is introduced. Using this concept, the $ (\alpha, \beta) $-Pythagorean fuzzy order of elements of groups in $ (\alpha, \beta) $-Pythagorean fuzzy subgroups is defined and examined various algebraic properties of it. A relation between $ (\alpha, \beta) $-Pythagorean fuzzy order of an element of a group in $ (\alpha, \beta) $-Pythagorean fuzzy subgroups and order of the group is established. The extension principle for $ (\alpha, \beta) $-Pythagorean fuzzy sets is introduced. The concept of $ (\alpha, \beta) $-Pythagorean fuzzy normalizer and $ (\alpha, \beta) $-Pythagorean fuzzy centralizer of $ (\alpha, \beta) $-Pythagorean fuzzy subgroups are developed. Further, $ (\alpha, \beta) $-Pythagorean fuzzy quotient group of an $ (\alpha, \beta) $-Pythagorean fuzzy subgroup is defined. Finally, an $ (\alpha, \beta) $-Pythagorean fuzzy version of Lagrange's theorem is proved. </abstract>

Crystallographic groups and flat manifolds from surface braid groups

Let M be a compact surface without boundary, and n≥2. We analyse the quotient group Bn(M)/Γ2(Pn(M)) of the surface braid group Bn(M) by the commutator subgroup Γ2(Pn(M)) of the pure braid group Pn(M). If M is different from the 2-sphere S2, we prove that Bn(M)/Γ2(Pn(M))≅Pn(M)/Γ2(Pn(M))⋊φSn, and that Bn(M)/Γ2(Pn(M)) is a crystallographic group if and only if M is orientable.If M is orientable, we prove a number of results regarding the structure of Bn(M)/Γ2(Pn(M)). We characterise the finite-order elements of this group, and we determine the conjugacy classes of these elements. We also show that there is a single conjugacy class of finite subgroups of Bn(M)/Γ2(Pn(M)) isomorphic either to Sn or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection Bn(M)/Γ2(Pn(M))⟶Sn is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups G˜n,g of Bn(M)/Γ2(Pn(M)) of dimension 2ng and whose holonomy group is the finite cyclic group of order n, and if Xn,g is a flat manifold whose fundamental group is G˜n,g, we prove that it is an orientable Kähler manifold that admits Anosov diffeomorphisms.

Topological groups without infinite precompact continuous homomorphic images

Let P be a property of topological groups. We say that a topological group G is MinAP moduloP if its image f[G] under each continuous homomorphism f:G→K from G to a compact group K has property P. When P is the property of being the trivial group, MinAP modulo P groups are precisely the minimally almost periodic groups of von Neumann and Wigner. We give a characterization of these new classes of groups for five properties P: finite, bounded, torsion, compact and connected. We show that these five classes are distinct and differ from the standard class of minimally almost periodic groups. We equip every Abelian group with a Hausdorff MinAP modulo finite group topology, thereby showing that, unlike the classical minimal almost periodicity, the property MinAP modulo finite imposes no restrictions whatsoever on the algebraic structure of the underlying group. At last but not least, we prove that the quotient group G/n(G) of every topological group G with respect to its von Neumann kernel n(G) is the categorical reflection of G in the class of maximally almost periodic (MAP) groups, and the natural quotient map plays the role of reflection homomorphism. As a consequence, this quotient group (G/n(G))+ equipped with its Bohr topology is topologically isomorphic to the image of G under the canonical homomorphism to its Bohr compactification.

Simply sm-factorizable (para)topological groups and their quotients

We say that a (para)topological group G is strongly submetrizable if it admits a coarser separable metrizable (para)topological group topology and is projectively strongly submetrizable if for each open neighborhood U of the identity in G, there is a closed invariant subgroup N contained in U such that the quotient (para)topological group G/N is strongly submetrizable. We show that a quotient group of a simply sm-factorizable ω-narrow topological abelian group can fail to be simply sm-factorizable. This answers a question posed by Arhangel'skii and the first listed author in 2018. If, however, the kernel of a quotient homomorphism is a bounded subgroup, then the homomorphism preserves simple sm-factorizability in the classes of topological and paratopological groups.We also prove that a regular (para)topological group G is simply sm-factorizable if and only if G is projectively strongly submetrizable and every continuous real-valued function on G is uniformly continuous on Gω, the P-modification of G. Making use of this fact we show that all weakly Lindelöf projectively strongly submetrizable paratopological groups and all weakly Lindelöf paratopological abelian groups are simply sm-factorizable. It is also established that every precompact paratopological group is simply sm-factorizable.

On Structural Properties of ξ -Complex Fuzzy Sets and Their Applications

Complex fuzzy sets are the novel extension of Zadeh’s fuzzy sets. In this paper, we comprise the introduction to the concept of ξ -complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of α , δ -cut sets of ξ -complex fuzzy sets and justify the representation of an ξ -complex fuzzy set as a union of nested intervals of these cut sets. We also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of ξ -complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an ξ -complex fuzzy set is ξ -complex fuzzy subgroup. Furthermore, we extend the idea of ξ -complex fuzzy normal subgroup to define the quotient group of a group G by this particular ξ -complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of G by a specific normal subgroup G A ξ .

Revisiting modular symmetry in magnetized torus and orbifold compactifications

We study the modular symmetry in $T^2$ and orbifold comfactifications with magnetic fluxes. There are $|M|$ zero-modes on $T^2$ with the magnetic flux $M$. Their wavefunctions as well as massive modes behave as modular forms of weight $1/2$ and represent the double covering group of $\Gamma \equiv SL(2,\mathbb{Z})$, $\widetilde{\Gamma} \equiv \widetilde{SL}(2,\mathbb{Z})$. Each wavefunction on $T^2$ with the magnetic flux $M$ transforms under $\widetilde{\Gamma}(2|M|)$, which is the normal subgroup of $\widetilde{SL}(2,\mathbb{Z})$. Then, $|M|$ zero-modes are representations of the quotient group $\widetilde{\Gamma}'_{2|M|} \equiv \widetilde{\Gamma}/\widetilde{\Gamma}(2|M|)$. We also study the modular symmetry on twisted and shifted orbifolds $T^2/\mathbb{Z}_N$. Wavefunctions are decomposed into smaller representations by eigenvalues of twist and shift. They provide us with reduction of reducible representations on $T^2$.

The Automorphism Group of Green Algebra of 9-Dimensional Taft Hopf Algebra

Let H3 be the 9-dimensional Taft Hopf algebra, let [Formula: see text] be the corresponding Green ring of H3, and let [Formula: see text] be the automorphism group of Green algebra [Formula: see text] over the real number field ℝ. We prove that the quotient group [Formula: see text] is isomorphic to the direct product of the dihedral group of order 12 and the cyclic group of order 2, where T1 is the isomorphism class which contains the identity map and is isomorphic to a group [Formula: see text] with multiplication given by [Formula: see text].

Amenable signatures, algebraic solutions and filtrations of the knot concordance group

It is known that each of the successive quotient groups of the grope and solvable filtrations of the knot concordance group has an infinite rank subgroup. The generating knots of these subgroups are constructed using iterated doubling operators. In this paper, for each of the successive quotients of the filtrations we give a new infinite rank subgroup which trivially intersects the previously known infinite rank subgroups. Instead of iterated doubling operators, the generating knots of these new subgroups are constructed using the notion of algebraic $n$-solutions, which was introduced by Cochran and Teichner. Moreover, for each slice knot $K$ whose Alexander polynomial has degree greater than 2, we construct the generating knots such that they have the same derived quotients and higher-order Alexander invariants up to a certain order. In the proof, we use an $L^2$-theoretic obstruction for a knot to being $n.5$-solvable given by Cha, which is based on $L^2$-theoretic techniques developed by Cha and Orr. We also generalize the notion of algebraic $n$-solutions to the notion of $R$-algebraic $n$-solutions where $R$ is either rationals or the field of $p$ elements for a prime $p$.

Generalized absolute values, ideals and homomorphisms in mixed lattice groups

A mixed lattice group is a generalization of a lattice ordered group. The theory of mixed lattice semigroups dates back to the 1970s, but the corresponding theory for groups and vector spaces has been relatively unexplored. In this paper we investigate the basic structure of mixed lattice groups, and study how some of the fundamental concepts in Riesz spaces and lattice ordered groups, such as the absolute value and other related ideas, can be extended to mixed lattice groups and mixed lattice vector spaces. We also investigate ideals and study the properties of mixed lattice group homomorphisms and quotient groups. Most of the results in this paper have their analogues in the theory of Riesz spaces.

On Small n-Hawaiian Loops

We define small n-Hawaiian loop as a special case of small pointed map, and study the group consisting of classes of small n-Hawaiian loops, $${\mathcal {H}}_n^s(X, x_0)$$ . As an example of spaces with non-trivial small 1-Hawaiian loops, we present 1st Hawaiian group of harmonic archipelago space as a special quotient group of 1st Hawaiian group of 1-dimensional Hawaiian earring. We define whisker topology on nth Hawaiian group which is Hausdorff if and only if $${\mathcal {H}}_n^s(X, x)$$ is trivial. We also prove that on metric spaces, null-homotopies can become small enough if and only if natural homomorphism maps $${\mathcal {H}}_n(X, x_0)$$ isomorphically on $$L_n(X, x_0)$$ if and only if $${\mathcal {H}}_n^s(X, x_0)$$ is trivial. Thus in spaces without non-trivial small n-Hawaiian loop, n-SLT paths transfer $${\mathcal {H}}_n$$ isomorphically along the points. We show that harmonic archipelago is an example with non-trivial small 1-Hawaiian loop and non-isomorphic 1st Hawaiian groups at two different points. Then we define n-SHLT paths which transfers nth Hawaiian group isomorphically along the points.