Quotient Group Research Articles

Unramified Brauer group of quotient spaces by finite groups

We give a general procedure to determine the unramified Brauer group of quotients of rational varieties by finite groups.

Distinguishing 2-knots admitting circle actions by fundamental groups

AbstractA 2-sphere embedded in the 4-sphere invariant under a circle action is called a branched twist spin. A branched twist spin is constructed from a 1-knot in the 3-sphere and a pair of coprime integers uniquely. In this paper, we study, for each pair of coprime integers, if two different 1-knots yield the same branched twist spin, and prove that such a pair of 1-knots does not exist in most cases. Fundamental groups of 3-orbifolds of cyclic type are obtained as quotient groups of the fundamental groups of the complements of branched twist spins. We use these groups for distinguishing branched twist spins.

Some characterizations of ω-balanced topological groups with a q-point

In this paper, we study some characterizations of q-spaces, strict q-spaces and strong q-spaces under ω-balanced topological groups. First we characterize a topological group G to be ω-balanced and a q-space whenever for each open neighborhood O of the identity in G, there is a countably compact invariant subgroup H which is of countable character in G, such that H ⊆ O and the canonical quotient mapping p : G → G/H is quasi-perfect and the quotient group G/H is metrizable. Secondly, we characterize G to be ω-balanced and a strict q-space, replacing the condition of being a q-space with an appropriate condition which is designed for the so-called “strict q-spaces”. Finally, we characterize G to be ω-balanced and a strong q-space, illustrating an additional condition to be replaced to that of q-space, or strict q-space. As applications of our three main results, we found new characterizations of the ω-narrow topological groups.

Isotropy group on some topological transformation group structures

This paper explores the topological properties of irresolute topological groups, their quotient maps, and the role of topology in normal subgroups. It provides a detailed analysis\linebreak using examples and counterexamples. The study focuses on the essential features of irresolute topological groups and their quotient groups, for understanding the topological aspects of isotropy groups. For a trans\-for\-ma\-tion group $(\mathsf{H}, \mathsf{Y}, \psi)$ and a point $y \in \mathsf{Y},$ the set \centerline{$\mathsf{H}_{y} = \{h \in \mathsf{H} \colon hy = y\}$} \noi consisting of elements of $\mathsf{H}$ that fix $y$, is called the isotropy group at $y$. The paper highlights the distinct topological characteristics of isotropy groups in transformation group structure. It demonstrates that if $(\mathsf{H}, \mathsf{Y}, \psi)$ is an Irr$^{*}$-topological transformation group, then $( \mathsf{H}/ \mathop{Ker} \psi, \mathsf{Y}, \overline{\psi})$ forms an effective Irr$^{*}$-topological transformation group. By investigating both irresolute topological groups and isotropy groups, the study provides a clear understanding of their topological features. This research improves our understanding of these groups by offering clear examples and counterexamples, leading to a thorough conclusion about their different topological features.

Enumeration of Spin-Space Groups: Toward a Complete Description of Symmetries of Magnetic Orders

Symmetries of three-dimensional periodic scalar fields are described by 230 space groups (SGs). Symmetries of three-dimensional periodic (pseudo)vector fields, however, are described by the spin-space groups (SSGs), which were initially used to describe the symmetries of magnetic orders. In SSGs, the real-space and spin degrees of freedom are unlocked in the sense that an operation could have different spatial and spin rotations. SSGs give a complete symmetry description of magnetic structures and have natural applications in the band theory of itinerary electrons in magnetically ordered systems with weak spin-orbit coupling. , a concept raised recently that belongs to the symmetry-compensated collinear magnetic orders but has nonrelativistic spin plitting, is well described by SSGs. Because of the vast number and complicated group structures, SSGs have not yet been systematically enumerated. In this work, we exhaust SSGs based on the invariant subgroups of SGs, with spin operations constructed from three-dimensional (3D) real representations of the quotient groups for the invariant subgroups. For collinear and coplanar magnetic orders, the spin operations can be reduced into lower-dimensional real representations. As the number of SSGs is infinite, we consider only SSGs that describe magnetic unit cells up to 12 times crystal unit cells. We obtain 157 289 noncoplanar, 24 788 coplanar-noncollinear, and 1421 collinear SSGs. The enumerated SSGs are stored in an online database with a user-friendly interface. We develop an algorithm to identify SSGs for realistic materials and find SSGs for 1626 magnetic materials. We also discuss several potential applications of SSGs, including the representation theory, topological states protected by SSGs, structures of spin textures, and refinement of magnetic neutron diffraction patterns using SSGs. Our results serve as a solid starting point for further studies of symmetry and topology in magnetically ordered materials. Published by the American Physical Society 2024

Comparative proteomic analyses of potato leaves from field-grown plants grown under extremely long days

There are limited molecular data and few biomarkers available for studies of field-grown plants, especially for plants grown during extremely long days. In this study we present quantitative proteomics data from 3 years of field trials on potato, conducted in northern and southern Sweden and analyze over 3000 proteins per year of the study and complement the proteomic analysis with metabolomic and transcriptomic analyses. Small but consistent differences linked to the longer days (an average of four more hours of light per day) in northern Sweden (20 h light/day) compared to southern Sweden can be observed, with a high correlation between the mRNA determined by RNA-seq and protein abundances. The majority of the proteins with differential abundances between northern and southern Sweden could be divided into three groups: metabolic enzymes (especially GABA metabolism), proteins involved in redox metabolism, and hydrolytic enzymes. The observed differences in metabolic enzyme abundances corresponded well with untargeted metabolite data determined by GC and LC mass-spectrometry. We also analyzed differences in protein abundance between potato varieties that performed relatively well in northern Sweden in terms of yield with those that performed relatively less well. This comparison indicates that the proteins with higher abundance in the high-yield quotient group are more anabolic in their character, whereas the proteins with lower abundance are more catabolic. Our results create a base of information about potato “field-omics” for improved understanding of physiological and molecular processes in field-grown plants, and our data indicate that the potato plant is not generally stressed by extremely long days.

Sums of squares in function fields over henselian discretely valued fields

Let n∈N and let K be a field with a henselian discrete valuation of rank n with hereditarily euclidean residue field. Let F/K be a function field in one variable. It is known that every sum of squares is a sum of 3 squares. We determine the order of the group of nonzero sums of 3 squares modulo sums of 2 squares in F in terms of equivalence classes of certain discrete valuations of F of rank at most n. In the case of function fields of hyperelliptic curves of genus g, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by 2n(g+1). We show that this bound is optimal. Moreover, in the case where n=1, we show that if F/K is a hyperelliptic function field such that the order of this quotient group is 2g+1, then F is nonreal.

Partition functions for U(1) vectors and phases of scalar QED in AdS

We extend the computation of one-loop partition function in AdSd+1 using the method in [23] and [24] for scalars and fermions to the case of U(1) vectors. This method utilizes the eigenfunctions of the AdS Laplacian for vectors. For finite temperature, the partition function is obtained by generalizing the eigenfunctions so that they are invariant under the quotient group action, which defines the thermal AdS spaces. The results obtained match with those available in the literature. As an application of these results, we then analyze phases of scalar QED theories at one-loop in d = 2, 3. We do this first as functions of AdS radius at zero temperature showing that the results reduce to those in flat space in the large AdS radius limit. Thereafter the phases are studied as a function of the scalar mass and temperature. We also derive effective potentials and study phases of the scalar QED theories with N scalars.

On the Isomorphism Problem for Central Extensions II

In this paper, we study the isomorphism problem for central extensions. More precisely, in some new situations, we provide necessary and sufficient conditions for two central extensions to be isomorphic. We investigate the case when the quotient group is simple or purely non-abelian. Furthermore, we characterize isomorphisms leaving the quotient group invariant. Finally, we deal with isomorphisms of central extensions where the kernel group and the quotient group are isomorphic.

Reidemeister–Schreier rewriting process for matching uniform signal constellations to quotient groups of arithmetic Fuchsian groups

Reidemeister–Schreier rewriting process for matching uniform signal constellations to quotient groups of arithmetic Fuchsian groups

Modular symmetry in magnetized T2g torus and orbifold models

We study the modular symmetry in magnetized T2g torus and orbifold models. The T2g torus has the modular symmetry Γg=Sp(2g,Z). The magnetic flux background breaks the modular symmetry to a certain normalizer Ng(H). We classify remaining modular symmetries by magnetic flux matrix types. Furthermore, we study the modular symmetry for wave functions on the magnetized T2g and certain orbifolds. It is found that wave functions on magnetized T2g as well as its orbifolds behave as the Siegel modular forms of weight 1/2 and N˜g(H,h), which is the metaplectic congruence subgroup of the double covering group of Ng(H), N˜g(H). Then, wave functions transform nontrivially under the quotient group, N˜g,h=N˜g(H)/N˜g(H,h), where the level h is related to the determinant of the magnetic flux matrix. Accordingly, the corresponding four-dimensional chiral fields also transform nontrivially under N˜g,h modular flavor transformation with modular weight −1/2. We also study concrete modular flavor symmetries of wave functions on magnetized T2g orbifolds. Published by the American Physical Society 2024

Growing trees from compact subgroups

We establish a new connection between local and large-scale structure in compactly generated totally disconnected locally compact (t.d.l.c.) groups G , finding a sufficient condition for G to have more than one end in terms of its compact subgroups. The condition actually results in an action of a quotient group G/N on a tree with faithful micro-supported action on the boundary, where N is compact, and is closely related to the Boolean algebra formed by the centralisers of the subgroups of G/N with open normaliser. As an application, we find a sufficient condition, given a one-ended t.d.l.c. group G , for all direct factors of open subgroups of G to be trivial or open.

Fixed Point Set and Equivariant Map of a S-Topological Transformation Group

The fixed point set and equivariant map of a S-topological transformation group is explored in this work. For any subset K of G, it is established that the fixed point set XK is clopen in X and for a free S-topological transformation group, it is proved that the fixed point set of K is equal to the fixed point set of closure and interior of the subgroup of G generated by K. Subsequently, it is proved that the map between STCG(X) and STCG’(X’) is a homomorphism under a Φ’- equivariant map. Also, it is proved that there is an isomorphism between the quotient topological groups and some basic properties of fixed point set of a S-topological transformation group are studied.

Further results on generalized cellular automata

Given a finite set A and a group homomorphism ϕ : H → G , a ϕ -cellular automaton is a function T : A G → A H that is continuous with respect to the prodiscrete topologies and ϕ -equivariant in the sense that h · T ( x ) = T ( ϕ ( h ) · x ) , for all x ∈ A G , h ∈ H , where · denotes the shift actions of G and H on A G and AH , respectively. When G = H and ϕ = id , the definition of id -cellular automata coincides with the classical definition of cellular automata. The purpose of this paper is to expand the theory of ϕ -cellular automata by focusing on the differences and similarities with their classical counterparts. After discussing some basic results, we introduce the following definition: a ϕ -cellular automaton T : A G → A H has the unique homomorphism property (UHP) if T is not ψ-equivariant for any group homomorphism ψ : H → G , ψ ≠ ϕ . We show that if the difference set Δ ( ϕ , ψ ) is infinite, then T is not ψ-equivariant; it follows that when G is torsion-free abelian, every non-constant T has the UHP. Furthermore, inspired by the theory of classical cellular automata, we study ϕ -cellular automata over quotient groups, as well as their restriction and induction to subgroups and supergroups, respectively.

Investigation of a neutrosophic group

We use a neutrosophic set, instead of an intuitionistic fuzzy because the neutrosophic set is more general, and it allows for independent and partial independent components μ (χ) , γ (χ) , ζ (χ), while in an intuitionistic fuzzy set, all components are totally dependent. In this article, we present and demonstrate the concept of neutrosophic invariant subgroups. We delve into the exploration of this notion to establish and study the neutrosophic quotient group. Further, we give the concept of a neutrosophic normal subgroup as a novel concept.

Zero sum subsequences and hidden subgroups

We propose a method for solving the hidden subgroup problem in nilpotent groups. The main idea is iteratively transforming the hidden subgroup to its images in the quotient groups by the members of a central series, eventually to its image in the commutative quotient of the original group, and then using an abelian hidden subgroup algorithm to determine this image. Knowing this image allows one to descend to a proper subgroup unless the hidden subgroup is the full group. The transformation relies on finding zero sum subsequences of sufficiently large sequences of vectors over finite prime fields. We present a new deterministic polynomial time algorithm for the latter problem in the case when the size of the field is constant. The consequence is a polynomial time exact quantum algorithm for the hidden subgroup problem in nilpotent groups having constant nilpotency class and whose order only have prime factors also bounded by a constant.

An Approach to Lagrange’s Theorem in Pythagorean Fuzzy Subgroups

The Pythagorean fuzzy environment is a modern way of depicting uncertainty. The concept of Pythagorean fuzzy semi-level subgroups of any group is described in this paper. The Pythagorean fuzzy order of an element in a Pythagorean fuzzy subgroup is introduced and established various algebraic attributes. The relation between the Pythagorean fuzzy order of an element of a group and the order of that group is established. The Pythagorean fuzzy normalizer and Pythagorean fuzzy centralizer of Pythagorean fuzzy subgroups are discussed. Further, the concept of Pythagorean fuzzy quotient group and the index of a Pythagorean fuzzy subgroup are defined. Finally, a framework is developed for proving Lagrange’s theorem in Pythagorean fuzzy subgroups.

$ (\epsilon, \delta) $-complex anti fuzzy subgroups and their applications

<abstract><p>The complex anti-fuzzy set (CAFS) is an extension of the traditional anti-fuzzy set with a wider range for membership function beyond real numbers to complex numbers with unit disc aims to address the uncertainty of data. The complex anti-fuzzy set is more significant because it provides two dimensional information and versatile representation of vagueness and ambiguity of data. In terms of the characteristics of complex anti-fuzzy sets, we proposed the concept of $ (\epsilon, \delta) $-CAFSs that offer a more comprehensive representation of the uncertainty of data than CAFSs by considering both the magnitude and phase of the membership functions and explain the $ \left(\epsilon, \delta \right) $-complex anti fuzzy subgroups (CAFS) in the context of CAFSs. Moreover, we showed that everyCAFSGis a $ (\epsilon, \delta) $-CAFSG. Also, we used this approach to define $ (\epsilon, \delta) $-complex anti-fuzzy(CAF) cosets and $ (\epsilon, \delta) $-CAF normal subgroups of a certain group as well as to investigate some of their algebraic properties. We elaborated the $ (\epsilon, \delta) $-CAFSG of the classical quotient group and demonstrated that the set of all $ (\epsilon, \delta) $-CAF cosets of such a particular CAFs normal subgroup formed a group. Furthermore, the index of $ \left(\epsilon, \delta \right) $-CAFSG was demonstrated and $ (\epsilon, \delta) $-complex anti fuzzification of Lagrange theorem corresponding to the Lagrange theorem of classical group theory was briefly examined.</p></abstract>

Groups of basic automorphisms of chaotic Cartan foliations with Eresmann connection

The purpose of the work is to study the groups of basic automorphisms of chaotic Cartan foliations with Ehresmann connection. Cartan foliations form a category where automorphisms preserve not only the foliation, but also its transverse Cartan geometry. The group of basic automorphisms of a foliation is the quotient group of the group of all automorphisms of this foliation by the normal subgroup of leaf automorphisms with respect to which each leaf is invariant. Cartan foliations include such wide classes of foliations as pseudo-Riemannian, Lorentzian, and foliations with transversal affine connection. No restrictions are imposed on the dimension of either the foliation or the foliated manifold. Compactness of the foliated manifold is not assumed. Methods. The proof of the structure theorem for chaotic Cartan foliations is based on the application of the foliated bundle construction, commonly used in the theory of foliations with transverse geometries. Results. The main result of this paper is the theorem stating that the group of basic automorphisms of any chaotic Cartan foliation with Ehresmann connection admits the structure of a Lie group and finding estimates for the dimension of this group. In particular, it is proved that if the set of closed leaves is countable, then the group of basic automorphisms of such a foliation is countable. Conclusion. In this paper, we prove a criterion according to which the chaoticity of a Cartan foliation of type (G, H) is equivalent to the chaoticity of a locally free action of the group H on the associated parallelizable manifold. Thus, the problem of the existence of chaos in Cartan foliations with Ehresmann connection reduces to the same problem for locally free actions of a Lie group on parallelizable manifolds.

Localization operator and Weyl transform on reduced Heisenberg group with multidimensional center

In this article, we study two different types of operators, the localization operator and Weyl transform, on the reduced Heisenberg group with multidimensional centre G. The group G is a quotient group of non-isotropic Heisenberg group with multidimensional centre Hm by its centre subgroup. Firstly, we define the localization operator using a wavelet transform on G and obtain the product formula for the localization operators. Next, we define the Weyl transform associated to the Wigner transform on G with the operator-valued symbol. Finally, we have shown that the Weyl transform is not only a bounded operator but also a compact operator when the operator-valued symbol is in Lp, 1 ≤ p ≤ 2, and it is an unbounded operator when p > 2.