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Even in the highest-level mathematics courses at undergraduate institutions, most students do not read mathematical research from professional journals. Such depth of content is often reserved for “independent studies” or summer REU's. In this article I describe a course that required students to read research papers, all of which centered on a single theme. The course covered advanced group theory topics while conveying the notion that there is a mathematical community that works together to advance mathematical knowledge. Using a single story as an integrative thread is an approach that can be adapted to any topic in mathematics. [ABSTRACT FROM AUTHOR]

Society is a composite of interacting people and groups. These groups play a significant role in maintaining social status, establishing group identity and social identity, and enforcing norms. As such, groups are essential for understanding human behavior. Nevertheless, the study of groups in everyday group life yields many diverse and sometimes contradicting theories of group behavior, and researchers tend to agree that we have yet to understand the emergence of groups out of aggregates of individuals. The current paper aims to shed new light on the convoluted interrelation between groups and individuals by focusing on individuals’ social identities and group categorization. It does so by exploring the dynamic nature of the self and its implications on identity and group membership, and introducing a framework recognizing the fluidity of groups and group categorization. Incorporating historical insights with contemporary theories, this paper argues for a flexible understanding of group dynamics that surpasses rigid in-group and out-group classifications, proposing instead that group affiliations exist along a continuum that reflects the ever-changing social landscape. [ABSTRACT FROM AUTHOR]

To what extent can hostility toward Greeks in Turkish politics be explained sociologically? This paper provides a fresh perspective by examining the value of group position theory in understanding negative perceptions of Greeks in Turkish politics between 1946 and 1960. It proposes that Turkish politicians' perceived threats to valued resources played a key role in their unfavorable portrayals of Greeks. To test the argument, the study scrutinizes all Turkish parliamentary speeches in this period that contain the word Greek (N = 652). Quantitative analysis demonstrates that speaking about border-related threats is a significant predictor of negative perceptions. Qualitative investigation details how politicians justified their negative comments. The paper concludes that the negative remarks about Greeks are a function of Turkish politicians' perceived threats to valued resources. [ABSTRACT FROM AUTHOR]

In this paper, we extend theories of group cognition and regulation to examine how regulation occurs as part of a learning ecosystem. We examine our instructional approach, Embedded Design, that uses Human-Computer Interaction (HCI) as a conduit for students to make sense of their sociotechnical world and the ways people design everyday interaction around technology. Our research questions centered on identifying (RQ1) the types of problems that require students' collective regulation,. Our population included learners aged 8-12 enrolled in a play-based after school HCI design club. Using micro-analytic techniques, we examined the HCI design processes of four teams over nine sessions and four learners over two years. We identified the most common problems students encountered, how regulation and negotiation played out across different levels of analysis, what types of learning occurred as participants worked to improve collaborative processes over time, and the role that technology played in the process. We end the paper by proposing a model of how nested collective knowledge building processes evolve over time and discuss the implications of our findings for K-12 HCI education. [ABSTRACT FROM AUTHOR]

González's paper (this issue) strives to deepen the field of psychoanalytic thinking and the growing body of theory bridging psyche and society, by reaching to group theory and fashioning an unconscious internal structure for the social links alive within us. His paper represents an incredibly ambitious, rich, and highly complex synthesis of aspects of classical theory, object relations, relational, neo-Kleinian, Winnicottian, intersectional, feminist, multiple group, link, and critical social psychoanalytic theories. In my discussion I will position González's ideas within the matrix of the Bionian psychic apparatus and field theories; followed by a response to his personal and clinical examples of the collective in the individual in the aftermath of the Trump election, and the challenges before us. [ABSTRACT FROM AUTHOR]

This paper is in continuation of our previous works. In this paper, we study solvable L-subgroups of an L-group and establish a level subset characterisation for the same. Then, this level subset characterisation has been used to describe solvability of L-subgroups with the help of the notions of normal and subinvariant series of L-subgroups. Moreover, the concept of supersolvable L-subgroups of an L-group has been introduced. It has been established that supersolvable L-groups are closed under the formation of subgroups. Also, commutator L-subgroup of a supersolvable L-subgroup is shown to be nilpotent. In the last, we extend Zassenhaus Lemma to L-setting and utilise it to establish a version of Schreier Refinement Theorem in L-group Theory. [ABSTRACT FROM AUTHOR]

This study contributes to the international literature on ethnic relations and multiculturalism by empirically examining the claim that ethnic diversity can undermine social cohesion. The specific focus is on analyzing the attitudes of Canadians toward different ethnic groups to assess the level of ethnic intolerance. Canada serves as an ideal case study for this investigation due to the long-standing federal policy of multiculturalism. The multicultural we is an ideal, composed of many us which are opposed to others, all of which are themselves composed of individuals manifesting a heterogeneity of attitudes and opinions. By studying out-group attitudes, the study seeks to assess the viability of multiculturalism in addressing the challenges posed by ethnic diversity. The study has three primary objectives: firstly, to examine if the distribution of ethnic intolerance varies across different ethnic groups in Canada; secondly, to investigate whether Canadians perceive the Canadian mosaic in a vertical (hierarchical) or horizontal (egalitarian) manner, drawing from group position theory; and finally, to explore whether ethnic groups in Canada hold their own subjective group position hierarchies distinct from the national average. To achieve these objectives, the paper conducts an original quantitative analysis of the 2014 Provincial Diversity Project Survey. [ABSTRACT FROM AUTHOR]

This paper offers research insights on how refugee women who experience domestic violence develop employment prospects. Guided by social capital theory and the concept of group social capital, the paper uses a qualitative approach to identify intrapersonal and interpersonal processes in a group intervention that assist women members to adjust their cognitive reasoning about their domestic violence experience and engage in behaviours that potentially enhance their employment prospects. The paper contributes to understanding how group processes can foster small wins that may enhance the employment prospects of this vulnerable group. [ABSTRACT FROM AUTHOR]

AbstractA subgroup X of a group G is called pronormal if X and Xg are conjugate in 〈X,Xg〉 for each element g of G. Pronormal subgroups play a relevant role in many problems of group theory; for instance, they arise naturally in the study of groups in which normality is a transitive relation, because a subgroup is normal if and only if it is subnormal and pronormal. Groups with only pronormal subgroups have been described by Kuzennyĭ and Subbotin (at least in the locally graded case) and the aim of this paper is to investigate the structure of uncountable locally graded groups of cardinality ℵ in which all subgroups of cardinality ℵ are pronormal. [ABSTRACT FROM AUTHOR]

AbstractLeibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance and ascendancy in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. Kawamoto studied ascendency in Lie algebras. In this paper we define subinvariance and asendency in Leibniz algebras and study their properties. It is shown that the signature results on subinvariance and ascendency in Lie algebras have analogs for Leibniz algebras. [ABSTRACT FROM AUTHOR]

The discourse of Iranian nationalism exhibits distinct attributes, including Aryanism, anti-Arabism, and a yearning to restore ancient Iran. Originating during the Qajar period and prevailing throughout the Pahlavi era, this discourse faced marginalization following the Islamic Revolution, remaining dormant for three decades. However, in the past decade, it has experienced a gradual resurgence and gained media representation, termed ‘Iranian neo-nationalism’ for reasons elucidated in the introduction. This study aims to investigate the factors underlying this resurgence. Utilizing the qualitative content analysis method and the theory of politicized group identity from a social psychology perspective, the paper contends that this re-emergence is grounded in enduring political, cultural, and global processes, as well as immediate cognitive and psychological processes. These intertwined processes reveal an ‘identity security dilemma’ in Iran. [ABSTRACT FROM AUTHOR]

AbstractWe define, and denote by anz(G), the average number of zeros of characters of a finite group G as the number of zeros in the character table of G dividing the number of irreducible characters of G. In this paper, we classify odd-order nonabelian groups G with anz(G)≤2. [ABSTRACT FROM AUTHOR]

AbstractThe conjugacy relation is an important tool in the study of group theory and now has been generalized to semigroups in several methods. In particular, the p-conjugacy, o-conjugacy, c-conjugacy, i-conjugacy, n-conjugacy and r-conjugacy in semigroups are introduced and investigated extensively and some problems related to these conjugacy relations are proposed. The aim of this paper is to continue the study of conjugacy relations in semigroups around some of these problems. We first point out that p-conjugacy in a semigroup that is embeddable in a group, is not necessarily transitive. Then we obtain a simple sufficient condition under which the o-conjugacy is the universal relation and prove that this condition is also necessary in some natural classes of semigroups. Finally, we explore some properties of r-conjugacy in restriction semigroups, which extends some results on i-conjugacy in inverse semigroups. Our results have answered or partially answered some problems raised by Araújo, Kinyon, Konieczny and Malheiro in [Proc.Roy.Soc.Edinburgh Sect.A 147, 1169–1214, 2017] and [J.Algebra 533, 142–173, 2019]. [ABSTRACT FROM AUTHOR]

In this paper, we reveal three classes of asylum seeker sentiment in the Australian population. The first group are pro-asylum seekers who think Australians should help refugees, that government agencies should not be turning back boats carrying people seeking asylum, that immigration levels are low, and that current government policy is too tough. The second group are anti-asylum seekers who strongly disagree with helping refugees and people seeking asylum, and support a tough border policy. A third group are pro-government policy but largely sympathetic to refugees and people seeking asylum. We explain the demography and attitudes of these three groups using the theory on border securitisation. The securitisation actions and discourse position people seeking asylum as a threat. This has received widespread public endorsement. This is an empirical reflection of over two decades of influential Australian government discourse of antipathy towards refugees and asylum seekers. Successive Australian governments have tied those seeking asylum who arrive by boat with strict border enforcement policies that target people seeking asylum as a security threat to be repelled from the Australian mainland. This largely negative discourse has supported the merits of tough border controls. However, this discursive campaign has not entirely eradicated sympathy towards refugees and asylum seekers. [ABSTRACT FROM AUTHOR]

Students often struggle with concepts from abstract algebra. Typical classes incorporate few ways to make the concepts concrete. Using a set of woven paper artifacts, this paper proposes a way to visualize and explore concepts (symmetries, groups, permutations, subgroups, etc.). The set of artifacts used to illustrate these concepts is derived from our investigation of open-work woven mats produced in several cultures in the South Pacific. The exemplars that will be shown present variations of the figure eight, and can be created using readily available materials and straightforward instructions. [ABSTRACT FROM PUBLISHER]

Given a construction f on groups, we say that a group G is f -realisable if there is a group H such that G ≅ f (H) , and completely f-realisable if there is a group H such that G ≅ f (H) and every subgroup of G is isomorphic to f (H 1) for some subgroup H 1 of H and vice versa. Denote by L (G) the absolute center of a group G, that is the set of elements of G fixed by all automorphisms of G. By using the structure of the automorphism group of a ZM-group, in this paper we prove that cyclic groups C N , N ∈ N * , are completely L-realisable. [ABSTRACT FROM AUTHOR]

Let G be a classical group of dimension d and let a = (a 1 , ... , a d) be differential indeterminates over a differential field F of characteristic zero with algebraically closed field of constants C. Further let A (a) be a generic element in the Lie algebra g (F 〈 a 〉) of G obtained from parametrizing a basis of g with the indeterminates a. It is known (cf. [5]) that the differential Galois group of y ′ = A (a) y over F 〈 a 〉 is G (C) . In this paper we construct a differential field extension L of F 〈 a 〉 such that the field of constants of L is C, the differential Galois group of y ′ = A (a) y over L is still the full group G (C) and A (a) is gauge equivalent over L to a matrix in normal form which we introduced in [13]. We also consider specializations of the coefficients of A (a) . [ABSTRACT FROM AUTHOR]

In this paper, all supercharacter theories of dicyclic groups T 4 n are described. These supercharacter theories are constructed by using the supercharacter theories of dihedral groups of order 4n which are classified by Lamar. [ABSTRACT FROM AUTHOR]

The paper considers very general multivariate modifications of Cramer-Lundberg risk model. The claims can be of different types and can arrive in groups. The groups arrival processes have constant intensities. The counting groups processes are dependent multivariate compound Poisson processes of Type I. We allow empty groups and show that in that case we can find stochastically equivalent Cramer-Lundberg model with non-empty groups. The investigated model generalizes the risk model with common shocks, the Poisson risk process of order k, the Poisson negative binomial, the Polya-Aeppli, the Polya-Aeppli of order k among others. All of them with one or more types of policies. The numerical characteristics, Cramer-Lundberg approximations, and probabilities of ruin are derived. During the paper, we show that the theory of these risk models intrinsically relates to the special types of integro differential equations. The probability solutions to such differential equations provide new insights, typically overseen from the standard point of view. [ABSTRACT FROM AUTHOR]

According to the orbit method, the construction of a unitary irreducible representation of a nilpotent Lie group requires a precise computation of some polarizing subalgebra subordinated to a linear functional in the linear dual of the corresponding Lie algebra. This important step is generally challenging from a computational viewpoint. In this paper, we provide an algorithmic approach to the construction of the well-known Vergne polarizing subalgebras. The algorithms presented in this paper are specifically designed so that they can be implemented in computer algebra systems. We also show there are instances where Vergne’s construction could be refined for the sake of efficiency. Finally, we adapt our refined procedure to free nilpotent finite-dimensional Lie algebras of step-two to obtain simple and precise descriptions of Vergne polarizing algebras corresponding to all linear functionals in a dense open subset of the linear dual of the corresponding Lie algebra. Also, a program written for Mathematica is presented at the end of the paper. [ABSTRACT FROM AUTHOR]

Five-axis machining of free-form surfaces has one major advantage over three-axis machining, i.e. a greater degree of flexibility in positioning the cutting tool relative to the surface. In five-axis machining of free-form surfaces there are three major phases in creating the tool path: (i) generation of the cutter contact (CC) points; (ii) formation of the cutter location (CL) points that define the path followed by the cutting tool reference point; and (iii) the creation of the specific machine tool part program (G-code file). In this paper, the free-form surface is defined as a triangular polyhedral mesh. The CC-points are created from the surface mesh definition employing a cutting plane technique. Additionally, the CC points are positioned based on an examination of several important factors: geometric constraints derived from the machine tool axis limits, gouging (undercutting) of the free-form surface by the cutting tool and collision of the tool with either the machining stock or machine. The CL points can then be generated along the resulting CC points by consideration of specific machining strategies, such as cusp height and a smooth change in tool posture. The critical issues addressed in this work concern the avoidance of machining problems (machine limit, collision, and gouging) in the tool-path generation phase. Therefore, this technique avoids inefficient five-axis machining practices by automatically creating and verifying a feasible tool-path prior to the actual metal cutting. [ABSTRACT FROM AUTHOR]

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an α G -partial Galois extension of S α G and a normal subgroup H of G, we prove that α G induces a unital partial action α G / H of G/H on the subalgebra of invariants S α H of S such that S α H is an α G / H -partial Galois extension of S α G . Second, assuming that G is abelian, we construct a commutative inverse semigroup T par (G , R) , whose elements are equivalence classes of α G -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between T par (G , R) / ρ and T(G, A), where ρ is a congruence on T par (G , R) and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of T par (G , R) reduces to the case where G is cyclic. The set of idempotents of T par (G , R) is also investigated. [ABSTRACT FROM AUTHOR]

Much has been written about the benefits to be derived by incorporating the history of mathematics into its teaching. Of the various ways to do this, the use of original sources is among the most thrilling for the insights and the challenges it offers for students and instructors alike. In this paper, we describe examples of one particular approach to address these challenges that preserves the thrill of the read. [ABSTRACT FROM PUBLISHER]

Let SVir be the well-known super-Virasoro algebras. In this paper, we first prove that any super-skewsymmetric super-biderivation of SVir is inner. Based on this, we show that every linear super-commuting map ψ on SVir is of the form ψ(x) = f(x)c, wherefis a linear function from SVir to ℂ mapping the odd part of SVir to zero, andcis the central charge of SVir. [ABSTRACT FROM PUBLISHER]

Strong flatness and condition (P) property of right S-acts over a monoid S have been the topic of many papers on homological classification of monoids. In this paper, we continue to investigate the monoids over which every right S-act satisfying condition (P) is strongly flat and we obtain some new classes of monoids. [ABSTRACT FROM AUTHOR]

An efficient identity-based encryption (IBE) scheme over lattice is proposed in this paper. Under the hardness of the learning with errors (LWE) problem, the proposed scheme is semantic secure against adaptive chosen identity and chosen plaintext attack in the standard model. To improve the efficiency of the lattice-based IBE scheme, unlike the identity string is encoded into a matrix by a group of public matrices in several known constructions, the identity string oflbits is encoded into a vector with the help ofvectors in this paper. With the help of this idea, we achieve the private key extraction of IBE scheme at the same lattice. Then, the public key of the proposed scheme only consists of onematrix andvectors, compared with that the public keys of the known lattice-based IBE schemes all consist as a group ofmatrices. Hence, the public key size of this scheme is shorter than that of the known constructions. [ABSTRACT FROM AUTHOR]

Let G be a group, let H be a subgroup of G and let Or(G) be the orbit category. In this paper we extend the definition of the relative group homology the- ories of the pair (G, H) defined by Adamson and Takasu to have coefficients in an Or(G)-module. There is a canonical comparison homomorphism defined by Cisneros- Molina and Arciniega-Nevrez from Takasu's theory to Adamson's theory. We give a necessary and sufficient condition on the subgroup H for which the comparison homomorphism is an isomorphism for all coefficients. We also use the Lück-Wiermann construction to introduce a long exact sequence for Adamson homology. Finally, we provide some examples of explicit computations for the comparison homomorphism. [ABSTRACT FROM AUTHOR]

Although cryptography is monumentally important in keeping our information safe and secure, this interdisciplinary field of study can also be used as a powerful teaching tool by putting mathematics in a dramatic and realistic setting. Cryptography also allows for a natural way to introduce topics such as modular arithmetic, matrix operations, and elementary group theory. This paper reports the results of a preliminary qualitative study of students' thinking and understanding of cryptography using both the SOLO taxonomy and open coding as they participated in task-based interviews. Students interviewed were assessed on how they made connections to various areas of mathematics through solving cryptography problems. Analyzing these interviews showed that students have a strong foundation in number-theoretic concepts such as divisibility and modular arithmetic. Also, students were able to use probability intuitively to make sense of and solve problems. Finally, participants' SOLO levels ranged from uni-structural to extended abstract, with the multi-structural level being the most common (three participants). This gives evidence to suggest that students should be given the opportunity to make mathematical connections early and often in their academic careers. Results indicate that although cryptography need not be a required course for mathematics majors, concepts from this field could be introduced in courses such as linear algebra, abstract algebra, number theory, and discrete mathematics. [ABSTRACT FROM AUTHOR]

IT has unarguably become a technology pillar for companies operating in highly competitive markets, characterised by shrinking margins. A typical example is the flexible manufacturing domain, where effective management of IT is the enabling factor of competitiveness. The challenge for these companies is achieving effective IT management at low costs. In this context, cloud computing is an extremely promising paradigm. However, unleashing this potential is not a trivial task, since it requires the ability to evaluate the pros and cons of a wide range of technologies and commercial offerings from a multitude of angles. In this paper, we propose an approach based on fuzzy set theory for enabling a group of companies to select the most convenient cloud offering available on the market. The approach is presented with respect to a real case study in the flexible manufacturing domain, specifically of agro-food industries. [ABSTRACT FROM AUTHOR]

In this paper, we determine the structure of certain algebraic morphisms and isometries of the spaceof allcomplex positive definite matrices. In the case, we describe all continuous Jordan triple endomorphisms ofwhich are continuous mapssatisfyingIt has recently been discovered that surjective isometries of certain substructures of groups equipped with metrics which are in a way compatible with the group operations have algebraic properties that relate them rather closely to Jordan triple morphisms. This makes us possible to use our structural results to describe all surjective isometries ofthat correspond to any member of a large class of metrics generalizing the geodesic distance in the natural Riemannian structure on. Finally, we determine the isometry group ofrelative to a very recently introduced metric that originates from the divergence called Stein’s loss. [ABSTRACT FROM AUTHOR]

In this paper, we shall give a refinement of the geometric realization of Lusztig’s algebra f given by Lusztig. This realization gives a geometric interpretation of the decomposition of f. As an application, we shall give geometric realizations of Lusztig’s symmetries on the whole quantum group. [ABSTRACT FROM AUTHOR]

In this paper and a sequel, we study a group which is the quotient of a free product of groups by the normal closure of a single word that is contained in a subgroup which has the form of a free product of two cyclic groups. We use known properties of generalized triangle groups, together with detailed analysis of pictures and of words in free monoids, to prove a number of results such as a Freiheitssatz and the existence of Mayer-Vietoris sequences for such groups under suitable hypotheses. The results generalize those in an earlier article of the second author and Shwartz. [ABSTRACT FROM AUTHOR]

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called ℨ-S-semipermutable if H permutes with every Sylow p-subgroup of G in ℨ for all p∉π(H); H is said to be ℨ-S-seminormal if it is normalized by every Sylow p-subgroup of G in ℨ for all p∉π(H). The main aim of this paper is to characterize the ℨ-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in ℨ are ℨ-S-semipermutable in G and the ℨ-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in ℨ are ℨ-S-seminormal in G. [ABSTRACT FROM AUTHOR]

A conjugacy class C of a finite group G is a sign conjugacy class if every irreducible character of G takes value 0,1 or −1 on C. In this paper, we classify the sign conjugacy classes of alternating groups. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of establishing exact formulas for calculating the maximum and minimum ranks of matrix expressions or matrix equalities that involve two matrices and their generalized inverses of matrices. For a pair of matricesAandBsuch that the productABis defined, two fundamental matrix expressions or matrix equalities composed by their generalized inverses are given byand, whereandare the-inverses ofAandB, respectively. Recently, the present author established a group of exact formulas for calculating the max–min ranks of the products, and derived many fundamental algebraic properties offrom the rank formulas. As a continuation, we study in this paper the ranks of the multiple matrix products,,,,, as well as the matrix differences,,,. All these kinds of problems are in fact to characterize algebraic properties of certain matrix-valued functions and we need to utilize a variety of known matrix rank formulas during this tedious work. [ABSTRACT FROM PUBLISHER]

In this paper, we apply a group theory method to derive the generators of the Lie algebra for a class of (k+1)th order difference equations. We then find solutions to the difference equations and study periodicity by giving sufficient conditions for existence of certain periodicities. We further look at the asymptotic behaviour of solutions satisfying certain properties. Our results generalise some known results in the literature. [ABSTRACT FROM AUTHOR]

It is known that normal affine semigroup rings are of finite F-representation type. In this paper, we prove that non-normal affine semigroup rings are also of finite F-representation type. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the orderm k-automorphisms ofSL(2,k). We first characterize the forms that orderm k-automorphisms ofSL(2,k) take and then we find simple conditions on matricesAandB, involving eigenvalues and the field that the entries ofAandBlie in, that are equivalent to isomorphy between the orderm k-automorphismsInnAandInnB. We examine the number of isomorphy classes and conclude with examples for selected fields. [ABSTRACT FROM PUBLISHER]

A group G is called mixed identity-free if for every n ∈ ℕ and every w ∈ G ∗ F n , there exists a homomorphism φ : G ∗ F n → G such that φ is the identity on G and φ (w) is nontrivial. In this paper, we make a modification to the construction of elementary amenable lacunary hyperbolic groups provided by Ol'shanskii et al. to produce finitely generated elementary amenable groups which are mixed identity-free. As a byproduct of this construction, we also obtain locally finite p-groups which are mixed identity-free. [ABSTRACT FROM AUTHOR]

Given a finite groupG, we denote byΔ(G) the commuting graph ofGwhich is defined as follows: the vertex set isGand two distinct verticesxandyare joined by an edge if and only ifxy = yx. Clearly,Δ(G) is always connected for any groupG. We denote byκ(G) the number of spanning trees ofΔ(G). In the present paper, among other results, we first obtain the valueκ(G) for some specific groupsG, such as Frobenius groups, Dihedral groups, AC-groups, etc. Next, we characterize the alternating groupA5, in the class of nonsolvable groups through its tree-numberκ(A5). Finally, we classify the finite groups for which the power graph and the commuting graph coincide. [ABSTRACT FROM PUBLISHER]

The notion of a quantum family of maps has been introduced in the framework ofC -algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a state). In this paper, we define a quantum family of homomorphisms of locally compact quantum groups. Roughly speaking, we show that such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras, is introduced and the algebraic counterpart of the aforementioned result is proved. Moreover, we show that a quantum family of homomorphisms of Hopf algebras is consistent with the counits and coinverses of the given Hopf algebras. We compare our concept withweak coactionsintroduced by Andruskiewitsch and we apply it to the analysis of adjoint coaction. [ABSTRACT FROM AUTHOR]

This paper provides the first steps in classifying the finite solvable groups having Property A, which is a property involving abelian normal subgroups. We see that this classification is reduced to classifying the solvableChermak–Delgado simple groups, which the author defines. The notion of “Chermak–Delgado simple,” or “CD-simple” for short, is a generalization of simple groups through the Chermak–Delgado lattice. The author completes a classification of Chermak–Delgado simple groups under certain restrictions on the primes involved in the group order. [ABSTRACT FROM PUBLISHER]

LetSbe a semigroup and 𝔽 be a field. For an idealJof the semigroup algebra 𝔽[S] ofSover 𝔽, let ϱJdenote the restriction (toS) of the congruence on 𝔽[S] defined by the idealJ. A semigroupSis called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β ofS. In this paper we show that ifSis a semilattice or a rectangular band then φ{S; 𝔽}: J → ϱJis a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (ℬS; ○ ) if and only ifSis a permutable semigroup. [ABSTRACT FROM PUBLISHER]

Students in an introductory abstract algebra course often struggle with the basic concept of a group. I present an art project I assigned to my students to help address this challenge in which they were asked to visualize a group in an artistic medium of their choice. The resulting art was often pleasing to my eye and revealed the aspects of the structure of the depicted group. [ABSTRACT FROM AUTHOR]

Using an autoethnographic approach and drawing on principles of reflective practice, this paper illustrates how a long-standing professional involvement in community education evolved into a personal search for meaning that included engagement with the concept of spirituality. It re-views typologies of community education developed in the UK in the 1980s and argues that, when they are viewed holistically, a spiritual dimension is implicit in them. The paper suggests that spirituality needs to be addressed as a dimension of educational practice and research in order to keep pace with growing popular interests and emergent thinking in the natural sciences. Two roads diverged in a wood, and I— I took the one less travelled by, And that has made all the difference. (Frost 1920) [ABSTRACT FROM AUTHOR]

This paper develops a neglected area in psychoanalytic theorizing: namely the collective aspects of individual subjectivity in the one-to-many object relationship. Using group theory allows us to describe this point of articulation between the individual and the group, allowing a more nuanced theoretical development of the social link in psychoanalysis. Subjectivity is inter-subjectivity in a radical way that extends beyond a singular one and singular other to pluralities of differences. We are both particles and waves; our subjectivity has a double provenance. This makes subjectivity inherently dynamic, inherently unstable, inherently dependent on groups of others, just as groups of others depend on the individuals that comprise them. [ABSTRACT FROM AUTHOR]

In this paper we investigate the group of L -homeomorphisms of an L -topological space. If there exists an L -topology δ on a set X such that the group of L -homeomorphisms of the L -topological space (X , δ) is a permuation group K on X , then K is L f -representable on X. It is proved that the direct sum of finite L f -representable permutation groups is L f -representable on X. We also investigate L f -reprsentability of some cyclic subgroups of the group S X . [ABSTRACT FROM AUTHOR]

Adopting the notion of a ( k ∗ , q) -quasi-coincidence of a fuzzy point with a fuzzy set, the idea of an (∈,∈∨(k∗, qk))-antifuzzy left (right) ideal, (∈,∈∨(k∗, qk))-antifuzzy ideal and (∈,∈∨(k∗, qk))-antifuzzy (generalized) bi-ideal in ordered semigroups are proposed, that are the generalization of the idea of an antifuzzy left (right) ideal, antifuzzy ideal and antifuzzy (generalized) bi-ideal in ordered semigroups and a few fascinating characterizations are obtained. In this paper, we tend to focus to suggest a connection between standard generalized bi-ideals and (∈,∈∨(k∗, qk))-antifuzzy generalized bi-ideals. In addition, different classes of regular ordered semigroups are characterized by the attributes of this new idea. Finally, the ( k ∗ , k) -lower part of an (∈,∈∨(k∗, qk))-antifuzzy generalized bi-ideal is outlined and a few characterizations are mentioned. [ABSTRACT FROM AUTHOR]

Do interest groups and corporate lobbyists tend to support or oppose bills moving through Congress? Do they oppose legislation in early, malleable stages, such as in committees, or are they more likely to oppose legislation in later stages when majorities of lawmakers are pushing bills towards final votes? And can pressure form legislators force lobbyists to support bills inconsistent with their members' interests? In this paper, I use data developed by the MapLight organisation on position taking by lobbyists lobbying the US Congress over ten years to find answers. Analysing a model built on theories of interest group influence, legislative behaviour, and the separation of powers, I find that lobbyists strategically choose their positions on bills for a variety of reasons, often prioritizing their need to please legislative allies over the need to faithfully represent interest group members or clients. The results raises serious concerns about representation by lobbyists. [ABSTRACT FROM AUTHOR]

In [Tărnăuceanu, M. (2015). Finite groups with a certain number of cyclic subgroups. Amer. Math. Monthly. 122:275–276], Tărnăuceanu described the finite groups G having exactly cyclic subgroups. In [Belshoff, R., Dillstrom, J., Reid, L. Finite groups with a prescribed number of cyclic subgroups. To appear in Communications in Algebra], the authors used elementary methods to completely characterize those finite groups G having exactly cyclic subgroups for Δ = 2, 3, 4 and 5. In this paper, we prove that for any Δ > 0 if G has exactly cyclic subgroups, then and therefore the number of such G is finite. We then use the computer program GAP to find all G with exactly cyclic subgroups for. [ABSTRACT FROM AUTHOR]