An integral domain D is called a finite factorization domain (FFD) if every nonzero nonunit element of D has only finitely many non-associate divisors. In 1998, for an integral domain D and a cancellative torsion-free monoid S such that each nonzero element of its quotient group is of type ( 0 , 0 , … ) , Kim proved that the monoid domain D[S] is an FFD if and only if D is an FFD and S is an FFM. However, it is still open whether a monoid algebra K[S] is an FFD provided that S is a reduced FFM. In this paper, we show that a Puiseux algebra K[S] is an FFD if and only if S is an FFM, when K is a finitely generated field of characteristic 0. This would provide a large class of one-dimensional monoid algebras with finite factorization property. We also prove that every generalized cyclotomic polynomial has the finite factorization property in K[S] where S is a reduced FFM and K is an arbitrary field of characteristic 0.

In this paper, a new type of soft group called the soft symmetric difference group (SSD-group) is introduced and systematically developed. This structure is constructed by integrating soft set theory with group theory through the symmetric difference operation and set inclusion. Fundamental concepts such as characteristic soft symmetric difference groups, soft symmetric difference subgroups, normal soft symmetric difference subgroups, soft normalizers, and soft cosets are defined, and their essential algebraic properties are investigated. Several characterizations of soft normality are also established through these concepts. Various axiomatic results are obtained, providing necessary and sufficient conditions for a soft set to form an SSD-group. Furthermore, soft quotient (factor) groups of SSD-groups are introduced and their structural properties are examined in detail. The relationship between SSD-group theory and classical group theory is also established through several corresponding concepts. Illustrative examples are provided to demonstrate the applicability and internal consistency of the proposed framework. Overall, the results obtained in this study extend existing soft group structures and contribute to the development of algebraic theory within the context of soft sets, while also providing a foundation for further generalizations to other algebraic frameworks such as semigroups, rings, and fields.

Let H H be a (multiplicatively written) monoid. The family P fin , 1 ( H ) \mathcal {P}_{\text {fin},1}(H) of finite subsets of H H containing the identity element is itself a monoid when endowed with setwise multiplication induced by H H . Tringali and Yan proved that two monoids H 1 H_1 and H 2 H_2 contained in a special class of commutative, cancellative monoids are isomorphic if and only if P fin , 1 ( H 1 ) \mathcal {P}_{\text {fin},1}(H_1) and P fin , 1 ( H 2 ) \mathcal {P}_{\text {fin},1}(H_2) are. Moreover, they raised the question whether the same holds in the general setting of cancellative monoids. We show that if H 1 H_1 and H 2 H_2 are (commutative) valuation monoids with trivial unit groups and isomorphic quotient groups, then P fin , 1 ( H 1 ) ≃ P fin , 1 ( H 2 ) \mathcal {P}_{\text {fin},1}(H_1)\simeq \mathcal {P}_{\text {fin},1}(H_2) . This provides a negative answer to Tringali and Yans question already within the class of valuation submonoids of the additive group Z 2 \mathbb {Z}^2 .

Let M be a closed surface, q ≥ 2 and n ≥ 2 . In this paper, we analyze the Coxeter-type quotient group B n ( M ) ( q ) of the surface braid group B n ( M ) by the normal closure of the element σ 1 q , where σ 1 is the standard Artin generator of the braid group B n . Also, we study the Coxeter-type quotient groups obtained by taking the quotient of B n ( M ) by the commutator subgroup of the respective pure braid group P n ( M ) , P n ( M ) and adding the relation σ 1 q = 1 , when M is a closed orientable surface or the disk.

This study presents a comprehensive characterization of the symmetry structures of tilings on a Klein bottle, arising from tilings of the Euclidean plane with crystallographic symmetry groups {\cal G} containing a subgroup {\cal L} of type pg. The investigation focuses on determining the isometries in the normalizer group {N_{\cal G}}\left({\cal L} \right) through diagrams of isometries, utilizing subgroup relationships among plane groups to streamline computations. A key result reveals that the quotient group {N_{\cal G}}\left({\cal L} \right)/{\cal L} can be decomposed as a product of cyclic and dihedral groups, with its order depending solely on the power of the generating translation of {\cal L} in the direction of the glide reflection axes.

Abstract We obtain the modular automorphism group of any quotient modular curve of level , with . In particular, we obtain some unexpected automorphisms of order 3 that appear for the quotient modular curves when the Atkin–Lehner involution belongs to the quotient modular group. We also prove that such automorphisms are not necessarily defined over . As a consequence of these results, we obtain the full automorphism group of the quotient modular curve , for sufficiently large .

Let [Formula: see text] be a group, and let [Formula: see text] be an automorphism of [Formula: see text]. If [Formula: see text] then [Formula: see text] is said to be a commuting automorphism. The set of all such automorphisms is denoted by [Formula: see text]. This set does not necessarily form a subgroup of the automorphism group of [Formula: see text]. If [Formula: see text] does form a subgroup, then [Formula: see text] is said to be an [Formula: see text]-group. Let [Formula: see text] be a set of prime numbers. Define [Formula: see text] as the ring consisting of all rational numbers [Formula: see text], where [Formula: see text] and [Formula: see text] are coprime integers, and [Formula: see text] is a [Formula: see text]-number. The additive group of [Formula: see text] is denoted by [Formula: see text]. Now let [Formula: see text] and [Formula: see text] be two sets of primes, and let [Formula: see text] be a nonzero integer. Consider a generalized extraspecial [Formula: see text]-group [Formula: see text], defined as follows: [Formula: see text] Let [Formula: see text], where [Formula: see text] is a generalized extraspecial [Formula: see text]-group such that [Formula: see text] with [Formula: see text]. In this paper, we show that if [Formula: see text], then [Formula: see text] is a non-[Formula: see text]-group, and if [Formula: see text], then [Formula: see text] is an [Formula: see text]-group. As a consequence, we identify the conditions determining when the following groups are [Formula: see text]-groups or not: (i) the direct product of a generalized extraspecial [Formula: see text]-group and a free abelian group with finite rank, (ii) an extension of [Formula: see text] by a direct sum of finitely many copies of [Formula: see text], where [Formula: see text] is the additive group of rational numbers, (iii) an infinite Černikov [Formula: see text]-group which is non-abelian but each proper quotient group is abelian.

Abstract Let 𝔼 be the HNN-extension of a group 𝐵 with subgroups 𝐻 and 𝐾 associated by an isomorphism φ : H → K \varphi\colon H\to K . Suppose that 𝐻 and 𝐾 are normal in 𝐵 and ( H ∩ K ) ⁢ φ = H ∩ K (H\cap K)\varphi=H\cap K . Under these assumptions, we prove necessary and sufficient conditions for 𝔼 to be residually a 𝒞-group, where 𝒞 is a class of groups closed under taking subgroups, quotient groups, and unrestricted wreath products. Among other things, these conditions give new facts on the residual finiteness and the residual 𝑝-finiteness of the group 𝔼.

The aim of this contribution is to generalize a formula proved by Maurice de Gosson (de Gosson 2017) about weak values in the context of the phase-space formulation of Quantum Mechanics (Rundle and Everitt 2021), in order to express those weak values using tools coming from the harmonic analysis on Lie Groups (Faraut 2006). A general formula which enables us to compute weak values is proved, in which the integration on a Lie Group is substituted to the integration on phase-space, using Haar measures. Then this formula is applied to SU(2) and SO(3) and also to the quotient group G/H, where H is a normal subgroup of G.

Abstract We clarify and extend our earlier work [1, 2] where it was shown how to amend a scheme originally proposed by M. Gell-Mann to identify the three families of quarks and leptons of the Standard Model with the 48 spin- $$ \frac{1}{2} $$ 1 2 fermions of N = 8 supergravity that remain after absorption of eight Goldstinos, a scheme that in its original form is dynamically realized at the SU(3) × U(1) stationary point of gauged N = 8 supergravity. We explain how to deform and enlarge this symmetry at the kinematical level to the full Standard Model symmetry group SU(3) c ×SU(2) w ×U(1) Y , with the correct charge and chiral assignments for all fermions. The framework also leaves room for an extra U(1) B−L symmetry. This symmetry enhancement is achieved by embedding the Standard Model symmetries into (a quotient group of) K(E10), the ‘maximal compact subgroup’ of the maximal rank hyperbolic Kac-Moody symmetry E10, and an infinite prolongation of the SU(8) R-symmetry of N = 8 supergravity. This scheme, which is also supposed to encompass quantum gravity, cannot be realized within the framework of space-time based (quantum) field theory, but requires space-time and related geometrical concepts to be ‘emergent’. We critically review the main hypotheses underlying this construction.

The q-rung orthopair fuzzy set (q-ROFS) has been developed as an extension of the Pythagorean fuzzy set (PFS) to address ambiguity in various decision-making contexts. Group theory is a significant area of mathematics with numerous applications across various scientific fields. This paper examines q-rung orthopair fuzzy group theory, emphasizing the importance of q-ROFS and group theory. The concept of a q-rung orthopair fuzzy subgroup (q-ROFSG) is introduced, and its various algebraic properties are examined. A comprehensive investigation into q-rung orthopair fuzzy cosets (q-ROFCs) and q-rung orthopair fuzzy normal subgroups (q-ROFNSGs) has been conducted. The definitions of q-rung orthopair fuzzy homomorphism and isomorphism are presented. We extend the concept of the quotient group of a classical group V in relation to its normal subgroup U by introducing a q-ROFSG of V⁄U. The q-rung orthopair fuzzy variant of the three fundamental isomorphism theorems has been demonstrated.

This article explores the strategy of combining group theory and programming methods to study and solve the problem of irregular Rubik’s Cube. Group theory, as an important branch of algebra, is a powerful tool for studying symmetry and structural transformations. In the study of Rubik’s Cube, group theory is widely used to analyze various transformations and their inherent mathematical properties. Irregular Rubik’s Cube, compared to traditional standard Rubik’s Cube, has a more complex structure and transformation mode, which requires us to consider more subgroups, quotient groups, and their nested relationships when applying group theory.

The Article discusses neutrosophic left/right cosets, their properties, neutrosophic normal subgroups, and neutrosophic quotient groups with some theories and examples. The concept of Neutrosophic Groups was introduced by Kandasamy and Smarandache in their work in 2006 as part of a broader field of research in neutrosophy. Neutrosophic Groups are defined by classical NeutroAxioms according to the Neutrosophic Set theory, which has type-1, The paper explores various structures related to Neutrosophic Groups, including examples of neutrosophic groups, neutrosophic left/right cosets, neutrosophic normal subgroups, Neutrosophic Lagrange’s theorem, quotient groups, and their properties, several results of the theorems, and examples are demonstrated with Gt1 [I].

Abstract The fundamental group of a directed graph admits a natural sequence of quotient groups called ‐fundamental groups, and the ‐fundamental groups can capture properties of a directed graph that the fundamental group cannot capture. The fundamental group of a directed graph is related to path homology through the Hurewicz theorem. The magnitude‐path spectral sequence connects magnitude homology and path homology of a directed graph, and it may be thought of as a sequence of homology of a directed graph, including path homology. In this paper, we study relations of the ‐fundamental groups and the magnitude‐path spectral sequence through the Hurewicz theorem and the Seifert–van Kampen theorem.

Steroid metabolomics in neonatal populations is challenged by considerable physiological heterogeneity and technical variability, which complicate the interpretation and comparability of metabolite profiles. Effective normalization strategies are essential to ensure accurate data analysis in this context. We analyzed 24-hour urinary steroid profiles in a cohort of 50 neonates (including very preterm, late preterm, and full-term infants) using gas chromatography-mass spectrometry. Two normalization techniques were compared: probabilistic quotient normalization (PQN) and peer group normalization (PGN). Normalization performance was assessed via distribution metrics, correlation with anthropometric variables, and principal component analysis (PCA). PGN achieved superior distributional normalization, with 27 of 30 metabolites conforming to normality assumptions, compared to 21 using PQN. PGN also eliminated all significant correlations between steroid levels and anthropometric parameters, indicating effective reduction of physiological confounding. In contrast, PQN partially mitigated such associations but was less robust in handling high-abundance metabolites. PCA confirmed improved sample dispersion and group separation after normalization, with method-dependent differences in Scores Plot. Peer group normalization is a sophisticated approach to reducing physiological variability in neonatal steroid profiling. These observations lend further credence to PGN as a promising strategy for standardizing steroid metabolomics in the field of neonatology. Nevertheless, further validation is necessary to substantiate these findings.

In this manuscript, we introduce the concepts of ψ-bipolar-valued fuzzy set (ψ-BVFS), ψ-bipolar-valued fuzzy normal subgroup (ψ-BVFNSG), cut sets Mψ(υ,χ)(Cυ,χMψ)) of ψ-BVFS and ψ-BVFSG, and bipolar-valued fuzzy cosets (BVF cosets). Further, we explore some algebraic properties of newly defined ψ-BVFSG. In addition, we present some new results of homomorphism and quotient group of ψ-BVFSG. At the end, we provide an application of ψ-BVFS in decision making by using topsis method.

Abstract In this paper, we investigate equivariant quotients of the Real bordism spectrum's multiplicative norm by permutation summands. These quotients are of interest because of their close relationship with higher real ‐theories. We introduce new techniques for computing the equivariant homotopy groups of such quotients. As a new example, we examine the theories . These spectra serve as natural equivariant generalizations of connective integral Morava ‐theories. We provide a complete computation of the ‐localized slice spectral sequence of , where is the real sign representation of . To achieve this computation, we establish a correspondence between this localized slice spectral sequence and the ‐based Adams spectral sequence in the category of ‐modules. Furthermore, we provide a full computation of the ‐localized slice spectral sequence of the height‐4 theory . The ‐slice spectral sequence can be entirely recovered from this computation.

This article explores the strategy of combining group theory and programming methods to study and solve the problem of irregular Rubik’s Cube. Group theory, as an important branch of algebra, is a powerful tool for studying symmetry and structural transformations. In the study of Rubik’s Cube, group theory is widely used to analyze various transformations and their inherent mathematical properties. Irregular Rubik’s Cube, compared to traditional standard Rubik’s Cube, has a more complex structure and transformation mode, which requires us to consider more subgroups, quotient groups, and their nested relationships when applying group theory.

Some properties involving feeble compactness, III: (Weakly) compact-bounded topological groups

In recent works, the concept of proximal groups was introduced as a generalization of topological groups. In this paper, we define the concept of quotient proximal groups with the help of canonical mappings, which is not really the generalization of quotient topological groups, in general. We study some topological properties of this quotient proximal group. We prove some important theorems namely induced mapping theorem, first isomorphism theorem and second isomorphism theorem for proximal groups. We also study numerous applications of first isomorphism theorem. The study is supported by various examples.