The aim of this short research note is to present some results about a conjecture of Barker and Gelvin [J. Group Theory 25 (2022), pp. 973–995, Conjecture 1.5 ] claiming that any source algebra of a p p -block ( p p is a prime) of a finite group has the unit group containing a basis stabilized by the left and right actions of the defect group. We obtain some reduction theorems for the existence of stable unital basis in source algebras of p p -block algebras. Along the way we investigate this problem for the p p -blocks of some finite simple groups.

In our previous work, we introduced the concept of torsion angular bin strings (TABS), which is a discrete vector representation of a conformer's torsional angles. Through this discretization, conformational states can be counted, yielding an estimate of the upper limit of the expected conformational ensemble size (nTABS). Besides nTABS being used as a quantitative measure of molecular flexibility, TABS itself is a way of grouping the conformers of a molecule without picking thresholds. This feature of TABS is especially valuable, as selecting suitable thresholds for metrics such as heavy-atom root-mean-square deviation (RMSD) or shape Tanimoto is highly system-dependent and can thus be challenging when working with large sets of molecules. Here, we describe the update to the nTABS algorithm of the TABS package since the last release. In addition, we present a classification study of conformer ensembles by TABS and compare it to classifications by a shape Tanimoto metric. Scientific contribution In contrast to our previous implementation, which handled molecular topological symmetry by enumerating all possible combinations that were simply permutations of one another, the new implementation treats TABS as mathematical objects governed by group theory, specifically Burnside's Lemma. This approach requires substantially less code and delivers a notable improvement in computational speed. The study also builds upon our previously developed framework for categorization comparisons between TABS and heavy-atom RMSD. Here, we show the results of a similar comparison with a shape Tanimoto metric, which further support the hypothesis that TABS encode the shape of conformers in a meaningful way.

This study examines gender-based violence (GBV) against men in Uganda, with particular attention to its forms, causes, and consequences. Guided by hegemonic masculinity theory, muted group theory, plus power and control theory, the study explores how dominant gender norms, discursive exclusion, and non-physical strategies of domination shape men’s experiences of abuse and contribute to its underreporting in Uganda. Using a narrative inquiry approach, data were collected through semi-structured interviews with 15 Ugandan men aged 18 years and above who were married, divorced, or separated. Participants were selected using a combination of convenience and purposive sampling techniques. The data were analyzed thematically using MAXQDA 2020, resulting in three major themes: forms of GBV, causes of GBV, and consequences of GBV against men. Findings reveal that the primary causes of GBV against men include poverty, group pressure, adultery, family background, and female social support networks. The study further indicates that GBV against men manifests in multiple forms, including physical, sexual, verbal, psychological, and economic abuse. Consistent with power and control theory, many of these abuses were non-physical and aimed at exerting domination and control over male partners. The consequences of such violence included psychological trauma, financial losses, suicidal ideation, and marital separation. Hegemonic masculinity norms and muted group dynamics further emerged as key factors that silenced male victims, discouraged reporting, and limited institutional recognition of men as legitimate victims of GBV. These findings signify that GBV against men in Uganda is real, multifaceted, and underreported. It is driven by power dynamics and gender norms. A possible mitigation mechanism for GBV against men calls for inclusive policies, recognition, and responsive support systems nationally. The study recommends fair and inclusive hearings for all GBV victims, increased advocacy for men’s rights, expanded victim counseling services, and greater involvement of cultural and religious institutions in GBV prevention and response. To enhance awareness among policymakers and stakeholders, further research on GBV against men in Uganda and other global contexts is strongly recommended.

ABSTRACT Critiques of integration discourse highlight its top-down, state-driven nature and tendency to frame minorities as perpetual outsiders. This article examines how migrants construct racialised discourses of integration to position themselves as part of the white majority. Introducing discursive integration, it argues that through everyday talk, migrants actively perform belonging. This process operates via a dual strategy: Polish migrants construct the white English majority as a positive reference group, claiming racial similarity, while constructing non-white minorities as a negative reference group from whom they distance themselves. In doing so, they strategically subordinate Polishness to whiteness, discursively transforming from minority into majority members – they integrate through race. Drawing on qualitative research with Polish migrants in Bristol, England, the article uses reference group theory to show that participants compared themselves favourably to a normative white English majority while portraying non-white minorities as lazy, culturally incompatible, and threatening – embodying ‘failed integration’. Through this mechanism, Polish migrants discursively became part of the white majority. While situated in England, the findings offer a transferable model for understanding migrant integration in other racialised societies. The study argues that integration is a racialised project where claiming whiteness expands majority belonging – at the expense of more marginalised groups.

Abstract Paley-type partial difference sets and skew–Hadamard difference sets are classical objects in algebraic combinatorics, known for their rich connections with graph theory, coding theory, and group theory. In this paper, we explore new links between these combinatorial structures and group codes arising as ideals in finite group algebras. We construct such codes from difference sets and determine their dimensions in several cases. As an application of our links, we explicitly compute the full set of primitive central idempotents in certain abelian $$ p $$ p -group algebras, by employing the classical sets of quadratic residues and non-residues modulo $$ p $$ p , which are well-studied examples of difference and partial difference sets—we also obtain their dimensions and estimate their minimum weights.

Corrigendum to “Combined use of ‘‘two-step’’ and ‘‘field-theoretic’’ renormalization group theories for deterministic partial differential equations” [Phys. A 687 (2026) 131362

Conventional molecular graphs are often unable to reliably encode stereochemistry, especially for symmetric molecules, nontetrahedral centers, and transition states. To overcome this, we present StereoMolGraph, an open-source Python library implementing a stereochemistry-aware graph representation for molecules and condensed graphs of reactions. Our method uses permutation-invariant local stereodescriptors, grounded in group theory, to provide an extensible representation of chirality. Based on this we introduce methods allowing for robust comparison of stereoisomers, including the identification of enantiomerism and diastereomerism, and support fleeting stereochemistry in transition states. We demonstrate the library's utility for complex organic molecules and metal complexes and analysis of distinct chiral reaction pathways. With RDKit interoperability and visualization features, StereoMolGraph offers a practical and transparent tool for advanced stereochemically aware chemoinformatics workflows.

ABSTRACT Group theory has profoundly advanced physics and chemistry in systems with symmetries. Yet its use in structural engineering applications has not yet been fully explored beyond the aesthetics of symmetric designs. This work addresses two significant gaps that have limited the broader adoption of group‐theoretic methods in structural vibration analysis and clarifies their implications for structural design when multiple eigenvalues arise. First, a problem‐independent approach is presented with detailed derivations for constructing group representations for symmetric structures directly from the ‐invariance for structural vibration analysis. This method applies effectively to both dihedral groups and the higher‐order Platonic groups, including tetrahedral (), octahedral (), and icosahedral () symmetries. The method used in this work scales well with structural complexity and enables both explicit and canonical block diagonalizations. Second, this work provides a comprehensive guide to applying finite‐point‐group representations in structural vibration analysis and proves that the dimensions of irreducible representations determine eigenfrequency multiplicities. Although no optimization is performed in this work, this theoretical result has direct implications for structural optimization, resolving longstanding misconceptions about the coalescence of eigenvalues by showing that symmetry is the origin of repeated eigenfrequencies. The theoretical developments are validated on truss structures with dihedral and higher‐order symmetries, accurately predicting their eigenfrequency distributions.

Metal halide perovskites show exceptional potential for solar energy, thermoelectrics, catalysis, and other photochemical technologies, with performance rooted in electronic structure-driven properties. In ABX3 halide perovskites, localized and often aspherical local electron densities from B-site lone pairs or polarizable X- anions can distort the lattice. However, the links among electronic structure fluctuations and distortions like tilting of the BX6 octahedra and off-centering of the B-site from the center of its octahedron are not fully understood. Using group theory and ab initio molecular dynamics, we quantify how lone pairs, halide polarization, off-centering, and octahedral tilting interact in the cubic phase CsBBr3, with B = Pb, Sn, and Ge. We find that lone pair-induced off-centering and octahedral tilting are symmetry-decoupled. Instead, stereochemical lone pair expression of the B-site ion is correlated to octahedral tilting through the propensity of the B-site to form a transient, partial covalent bond with the surrounding halide ions that stiffens octahedral tilting modes. These results link local electronic asymmetry to structural fluctuations and suggest that dynamic modulation of electronic symmetry offers a pathway to control functional properties in halide perovskites.

In the realm of mental health discourse, considerable attention has traditionally been directed toward women’s well-being, resulting in a significant gap in understanding and addressing the unique challenges surrounding men’s mental health. This study aims to explore the factors contributing to the persistent stigma surrounding men's mental health in the Klang Valley, identify the communication barriers that discourage men from seeking help, and propose strategies to facilitate more supportive help-seeking behaviours. Employing a qualitative research design, this study conducted semi-structured interviews with ten male participants from the Klang Valley to capture rich, first-hand insights into their experiences and perceptions. Through thematic analysis, six major themes were identified: Toxic Masculinity Norms, Cognitive Barriers, Social Stigma Barriers, Structural and Financial Barriers, Narrative or Norms Shifts, and Supportive and Institutional Interventions. These themes reveal the complex interplay of cultural expectations, internalised stigma, and systemic barriers that shape men’s reluctance to seek mental health support. Framed through the lens of Muted Group Theory, the study highlights how men’s emotional expressions and help-seeking behaviours are frequently marginalised within dominant social narratives, often leaving them without the language, space, or societal permission to articulate their struggles. Participants in the study underscored the urgent need to reshape prevailing narratives by leveraging the influence of media, public figures, and grassroots community initiatives. They also stressed the importance of fostering greater support within families, workplaces, educational settings, and government institutions. This study contributes to public discourse by offering practical recommendations for media organisations, NGOs, and government agencies to design targeted mental health awareness strategies. It further underscores the need for continued research and sustained efforts to challenge harmful gender norms and promote help-seeking as a collective social responsibility.

Two-dimensional (2D) magnetic materials have attracted significant attention due to their unique properties and potential applications in spintronics. Using first-principles calculations and group theory analysis, we propose a novel two-dimensional chiral magnetic semiconductor ${\mathrm{FeZrCl}}_{6}$. The monolayer (ML) ${\mathrm{FeZrCl}}_{6}$ is dynamically stable and hosts a coplanar noncollinear 120\ifmmode^\circ\else\textdegree\fi{} antiferromagnetic ground state on a triangular lattice, characterized by vector-spin chirality. The group theory analysis reveals that this unique magnetic order activates the anomalous Hall effect (AHE) and magneto-optical Kerr effect (MOKE). During collective in-plane spin rotation, both the AHC and Kerr rotation angle exhibit $2\ensuremath{\pi}/3$ periodicity dependence on the azimuthal angle $\ensuremath{\varphi}$. Furthermore, out-of-plane spin canting generates a finite scalar-spin chirality, producing an emergent real space Berry phase and thereby inducing a chirality-driven topological Hall effect (THE) and topological magneto-optical Kerr effect (TMOKE) without spin-orbit coupling. These findings not only identify a promising material candidate for antiferromagnetic spintronics but also provide profound insights into the antiferromagnetic anomalous transport phenomena.

The full symmetry group is found for a system of nonlinear schrödinger equations describing the propagation of optical pulses in an isotropic media. It is shown, in particular, that the six-dimensional symmetry group found is composed of a scaling transformation and a rotation of the four-dimensional space, thereby proving that the symmetry group preserves the shape of solutions. A symmetry classification of one-dimensional subalgebras of the Lie algebra is performed and yields, in particular, the symmetry reduction to the most general system of equations satisfied by the solitary waves of the equation. Explicit soliton solutions of the equation are found by largely autonomous technics. The found solitons are used to recursively generate two new ones by means of two iterations using the symmetry group. Other properties of the system are also highlighted, as well as the possible connections between the theories of symmetry groups and Darboux transformations inspired by this study.

The dynamical state of cortical neural activity constrains the complexity of functions it can perform. A marginally stable dynamical state - called criticality - is thought to be beneficial for brain functions that require multiple time scales, broad dynamic range, and large information storage and transmission. A growing body of evidence suggests that criticality is a feature of healthy brain dynamics, but breaks down in certain brain disorders. Here we ask whether Parkinson’s disease incurs deviation from criticality compared to healthy controls. We analyze human resting state EEG activity in primary motor cortex. Parkinson’s patients exhibit prominent oscillatory brain activity in multiple frequency bands (low delta and high theta) that is not present in controls. Surprisingly, we find that these emergent oscillations are close to criticality, i.e., amplitude fluctuations with approximate temporal scale invariance. We compare traditional signatures of criticality and more principled measurements of proximity to criticality using our new approach based on information theory and temporal renormalization group theory. Our new approach and traditional methods agree, demonstrating that critical dynamics are not always associated with healthy states; Parkinson’s disease is associated with the emergence of near-critical oscillations in motor cortex.Author SummaryBrain function is thought to be optimal when its activity is near the border of order and chaos — a state called criticality. This state is thought to help the brain stay flexible and process information efficiently. We investigate whether Parkinson’s disease disrupts this balance like in other diseases and pathologies. Using resting EEG brain activity, we found that people with Parkinson’s show strong rhythmic signals not seen in healthy brains, and surprisingly these rhythms are also near the critical state. Using both established and new theoretical tools, we show that critical dynamics can accompany disease, suggesting that being closer to criticality is not always a sign of healthy brain function.

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.

The study of automorphisms of algebraic structures plays a central role in understanding their internal symmetries and structural behavior. This work investigates the automorphism structure induced by <i>finite subgroups within infinite groups</i>, with particular emphasis on how these automorphisms can be characterized, classified, and effectively utilized. The focus is on the interaction between a finite subgroup and the ambient infinite group, analyzing how subgroup-preserving automorphisms extend to global automorphisms and how constraints imposed by finiteness influence the overall automorphism group. Special attention is given to classes of infinite groups such as abelian, conjugacies, and certain residually finite groups where finite subgroup automorphisms exhibit rich and tractable behavior. Building on this theoretical framework, this work explores <i>applications to symmetric cryptography,</i> where algebraic symmetry and complexity are essential for secure cryptographic design. Finite subgroup automorphisms are shown to provide a promising foundation for constructing cryptographic primitives, including key generation mechanisms, conjugacy-based encryption schemes, and secure mixing transformations. The inherent difficulty of reversing automorphism actions in large infinite groups, combined with the controlled structure of finite subgroups, offers a balance between computational efficiency and cryptographic strength. In overall, this work bridges abstract group theory and practical cryptographic applications, demonstrating that finite subgroup automorphisms of infinite groups constitute a viable and mathematically robust framework for advancing symmetric cryptographic systems.

This JMCA Research Article extends the wonderfully beautiful JMCA Research Article “The Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups” ( see https://www.onlinescientificresearch.com/articles/the-strong-sylow-theorem-for-the-prime-p-in-simple-locally-finite-groups.pdf ) from 71 pages to 226 pages by adding Pages i to viii, -1 to -24, 72 to 266, and ix to x (see Page v). Pages i to -24 start with this Abstract and then gladly honour Prof. Otto H. Kegel’s beautiful paper [44], thereby explaining its relationship to its main result and to the 71-pages Research Article, and commenting on its final considerations. They then give an overview of the 226-pages Research Article, including some venues, the Front Cover and a Table of Contents, and explain in great detail the relationships of the 71-pages Research Article to JMCA and to the Ischia Group Theory (IGT) 2024 Conference and to its upcoming Proceedings, centred around the Talk given by the author at the Conference on April 11, the 120th birthday of Prof. Philip Hall. They then present a List of ten Open Issues most of which the author has solved already in yet unpublished work and describe the Issues of the De Luxe Edition. After remembering Rudi Schuricke’s magnificient Florentinische Nächte and the author’s wedding, they close with a Table of Contents of the Pages 1 to 266 which are contained in the May 2025 Issue. We nevertheless summarise the contents of Pages 72 to x. They start with a dedication to Helga (see Page 15), continue with the PowerPoint Presentation at IGT 2024, honour Ludvig M. Sylow, the discoverer and explorer of Sylow Theory, recall that Ischia was twice an Artist Colony, remember Philip Hall’s and Graham Higman’s very fundamental paper “On the p-Length of p-Soluble Groups and Reduction Theorems for Burnside’s Problem” thereby well explaining its relationship to the 71-pages Research Article, recall Some Historicals on Group Theory, in particular Philip Hall’s unpublished hand-written Lecture Notes on Group Theory and show these beautiful Lecture Notes completely on the 148 Pages 103 to 250, show a presentation of Wikipedia’s Classification of finite simple groups, remember Prof. Kegel and Prof. Hall at the famous Mathematisches Forschungsinstitut Oberwolfach (MFO), and finally as a great Back Cover refer to Philip Hall’s Archive at the London Mathematical Society and again to his Lecture Notes, and close by showing Who is Felix F. Flemisch & Another Abstract of the 71-pages Research Article.

Abstract We prove a central limit theorem (CLT) for the number of joint orbits of random tuples of commuting permutations. In the uniform sampling case this generalizes the classic CLT of Goncharov for the number of cycles of a single random permutation. We also consider the case where tuples are weighted by a factor other than one, per joint orbit. We view this as an analogue of the Ewens measure, for tuples of commuting permutations, where our CLT generalizes the CLT by Hansen. Our proof uses saddle point analysis, in a context related to the Hardy–Ramanujan asymptotics and the theorem of Meinardus, but concerns a multiple pole situation. The proof is written in a self-contained manner, and hopefully in a manner accessible to a wider audience. We also indicate several open directions of further study related to probability, combinatorics, number theory, an elusive theory of random commuting matrices, and perhaps also geometric group theory.

Abstract Trinh and Xue have proposed a startling conjecture on intersections of blocks of cyclotomic Hecke algebras occurring in modular representation theory of finite reductive groups. We prove this conjecture for all exceptional type groups apart from a few situations in type $$E_8$$ E 8 . We also give a conceptual proof in all cases where relative Weyl groups are cyclic. Furthermore, we propose several generalisations, to Suzuki and Ree groups, to non-rational Coxeter groups and even more generally to spetsial complex reflection groups, and confirm these in various cases.

Abstract We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is based on defining 2D coproducts through horizontal and vertical maps that satisfy compatibility and associativity conditions, enabling the consistent growth of vector spaces over lattice sites. We present several examples of 2D bialgebras, including group-like and Lie algebra-inspired constructions and a quasi-1D coproduct instance that is applicable to Taft-Hopf algebras and to quantum groups. The approach is further applied to the quantum group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:mi>s</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:math> , for which we construct 2D generalizations of its generators, analyze q -deformed singlet states, and derive a 2D R-matrix satisfying an intertwining relation in the semiclassical limit. Additionally, we show how tensor network states, particularly projected entangled pair states, naturally induce 2D coalgebra structures when supplemented with appropriate boundary conditions. Our results establish a local and algebraically consistent method to embed quantum group symmetries into higher-dimensional lattice systems, potentially connecting to the emerging theory of fusion 2-categories and categorical symmetries in quantum many-body physics.

The Hodgkin-Huxley equations have successfully described neuronal excitability for over seventy years, yet their mathematical structure remains empirically justified rather than theoretically explained. Why are gating variables bounded between 0 and 1? Why does sodium conductance depend on m3h rather than other combinations? Why does potassium depend on n4? Why do all rate functions contain exponential voltage dependencies? Why are the kinetics first-order? We demonstrate that these structural features arise naturally from three fundamental physical symmetries governing ion channel dynamics: the compactness of conformational state space, the scaling invariance of membrane conductance, and temporal translation invariance. Using Lie group theory, we show that these symmetries uniquely determine a mathematical structure in which: (1) gating variables are necessarily bounded, (2) voltage dependencies must be exponential, (3) exponents must be integers, and (4) kinetics must be first-order. The Hodgkin-Huxley equations, rather than mere empirical fits, emerge from fundamental symmetry principles. This framework establishes that neural electrophysiology obeys the same theoretical principles as modern physics, where symmetries constrain the form of dynamical equations. It further provides a principled basis for interpreting deviations from classical behavior as manifestations of additional symmetries or symmetry breaking.