Search

The original article can be found online at https://doi.org/10.1007/s00222-024-01257-1The publication of this article unfortunately contained mistakes. The style of the references was not correct. The corrected reference list is given below.The original article has been corrected.Publisher's NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.By Yuan Liu; Melanie Matchett Wood and David Zureick-BrownReported by Author; Author; Author [Extracted from the article]

In this paper, we study the Maxwell equations in the domain of outer-communication of the Schwarzschild black hole. We prove that if the middle components of the non-stationary solutions of the Maxwell equations verify a Morawetz-type estimate supported on a compact region in space around the trapped surface, then the components of the Maxwell fields decay uniformly in the entire exterior of the Schwarzschild black hole, including the event horizon. This is shown by making only use of Sobolev inequalities combined with energy estimates using the Maxwell equations directly. The proof does not pass through the scalar wave equation on the Schwarzschild black hole, does not need to decouple the middle components for the Maxwell fields, and would be in particular useful for the non-abelian case of the Yang–Mills equations where the decoupling of the middle components cannot occur. In fact, the estimates for the hereby argument are still valid for the Yang–Mills fields except for the Lie derivatives of the fields that are involved in the proof. [ABSTRACT FROM AUTHOR]

The enhanced power graph of a group G is a simple undirected graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we obtain the Laplacian spectrum of the enhanced power graph of certain non-abelian groups, viz. semidihedral, dihedral and generalized quaternion. Also, we obtained the metric dimension and the resolving polynomial of the enhanced power graphs of these groups. At the final part of this paper, we study the distant properties and the detour distant properties, namely: closure, interior, distance degree sequence, eccentric subgraph of the enhanced power graph of semidihedral group, dihedral group and generalized quaternion group, respectively. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to compute the probability Pr(G) that two elements of the group G, drawn at random with replacement, commute; that is, Pr(G) = Number of ordered pairs (x, y) ∈ G × G such that xy = yx/|G × G| = |G|² In particular, we compute Pr(G) for some groups such as the extraspecial groups of order p³, p prime, for the permutation groups G = Sn and G = An, n ≥ 5, for 10 non-abelian groups of order p4 and for simple groups of certain type. [ABSTRACT FROM AUTHOR]

Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research. [ABSTRACT FROM AUTHOR]

In this paper, we study the module-theoretic structure of classical Iwasawa modules. More precisely, for a finite abelian p-extension K/k of totally real fields and the cyclotomic Z p -extension K ∞ / K , we consider X K ∞ , S = Gal (M K ∞ , S / K ∞) where S is a finite set of places of k containing all ramifying places in K ∞ and archimedean places, and M K ∞ , S is the maximal abelian pro-p-extension of K ∞ unramified outside S. We give lower and upper bounds of the minimal numbers of generators and of relations of X K ∞ , S as a Z p [ [ Gal (K ∞ / k) ] ] -module, using the p-rank of Gal (K / k) . This result explains the complexity of X K ∞ , S as a Z p [ [ Gal (K ∞ / k) ] ] -module when the p-rank of Gal (K / k) is large. Moreover, we prove an analogous theorem in the setting that K/k is non-abelian. We also study the Iwasawa adjoint of X K ∞ , S , and the minus part of the unramified Iwasawa module for a CM-extension. In order to prove these theorems, we systematically study the minimal resolutions of X K ∞ , S . [ABSTRACT FROM AUTHOR]

The solubilizer of an element x of a profinite group G is the set of the elements y of G such that the subgroup of G generated by x and y is prosoluble. We propose the following conjecture: the solubilizer of x in G has positive Haar measure if and only if x centralizes 'almost all' the non-abelian chief factors of G. We reduce the proof of this conjecture to another conjecture concerning finite almost simple groups: there exists a positive c such that, for every finite simple group S and every (a , b) ∈ (Aut (S) ∖ { 1 }) × Aut (S) , the number of s is S such that 〈 a , b s 〉 is insoluble is at least c | S |. Work in progress by Fulman, Garzoni and Guralnick is leading to prove the conjecture when S is a simple group of Lie type. In this paper we prove the conjecture for alternating groups. [ABSTRACT FROM AUTHOR]

A finite non-Dedekind group G is called an N AC-group if all non-normal abelian subgroups are cyclic. In this paper, all finite N AC-groups will be characterized. Also, it will be shown that the center of non-nilpotent N AC-groups is cyclic. If N AC-group G has a non-abelian non-normal Sylow subgroup of odd order, then other Sylow subgroups of G are cyclic or of quaternion type. [ABSTRACT FROM AUTHOR]

The non-hyperelliptic connected components of the strata of translation surfaces are conjectured to be orbifold classifying spaces for some groups commensurable to some mapping class groups. The topological monodromy map of the non-hyperelliptic components projects naturally to the mapping class group of the underlying punctured surface and is an obvious candidate to test commensurability. In the present article, we prove that for the components |$\mathcal {H}(3,1)$| and |$\mathcal {H}^{nh}(4)$| in genus 3 the monodromy map fails to demonstrate the conjectured commensurability. In particular, building on the work of Wajnryb, we prove that the kernels of the monodromy maps for |$\mathcal {H}(3,1)$| and |$\mathcal {H}^{nh}(4)$| are large, as they contain a non-abelian free group of rank |$2$|⁠. [ABSTRACT FROM AUTHOR]

We investigate the power of counting in Group Isomorphism. We first leverage the count-free variant of the Weisfeiler–Leman Version I algorithm for groups [J. Brachter and P. Schweitzer, On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786] in tandem with bounded non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include: • Direct products of non-Abelian simple groups. • Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an O (1) -generated solvable group with solvability class poly log log n. This notably includes instances where the complement is an O (1) -generated nilpotent group. This problem was previously known to be in P [Y. Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in Proc. 28th Symp. Theoretical Aspects of Computer Science, Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp. 567–578, doi:10.4230/LIPIcs. STACS.2011.567], and the complexity was recently improved to L [J. A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler–Leman, in 24th Int. Symp. Fundamentals of Computation Theory, eds. H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18–21, 2023, Trier, Germany (Springer, 2023), pp. 234–247]. • Graphical groups of class 2 and exponent p > 2 [A. H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, J. Symb. Logic46(4) (1981) 781–788] arising from the CFI and twisted CFI graphs [J.-Y. Cai, M. Fürer and N.  Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica12(4) (1992) 389–410], respectively. In particular, our work improves upon previous results of Brachter and Schweitzer [On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786]. Notably, each of these families was previously known to be identified by the counting variant of the more powerful Weisfeiler–Leman Version II algorithm. We finally show that the q-ary count-free pebble game is unable to even distinguish Abelian groups. This extends the result of Grochow and Levet (ibid), who established the result in the case of q = 1. The general theme is that some counting appears necessary to place G r o u p I s o m o r p h i s m into P. [ABSTRACT FROM AUTHOR]

We show that if a group G acts geometrically by type-preserving automorphisms on a building, then G satisfies the weak Tits alternative, namely, that G is either virtually abelian or contains a non-abelian free group. [ABSTRACT FROM AUTHOR]

We show that two-dimensional Artin groups satisfy a strengthening of the Tits alternative: their subgroups either contain a non-abelian free group or are virtually free abelian of rank at most 2. When in addition the associated Coxeter group is hyperbolic, we answer in the affirmative a question of Wise on the subgroups generated by large powers of two elements: given any two elements a , b of a two-dimensional Artin group of hyperbolic type, there exists an integer n ≥ 1 such that a n and b n either commute or generate a non-abelian free subgroup. [ABSTRACT FROM AUTHOR]

In recent years, averaging operators on Lie algebras (also called embedding tensors in the physics literature) and associated tensor hierarchies have formed an efficient tool for constructing supergravity and higher gauge theories. A Lie algebra with an averaging operator is called an averaging Lie algebra. In the present paper, we introduce 2-term averaging L ∞ -algebras and give characterizations of some particular classes of such homotopy algebras. Next, we study non-abelian extensions of an averaging Lie algebra by another averaging Lie algebra. We define the second non-abelian cohomology group to classify the equivalence classes of such non-abelian extensions. Next, given a non-abelian extension of averaging Lie algebras, we show that the obstruction for a pair of averaging Lie algebra automorphisms to be inducible can be seen as the image of a suitable Wells map. Finally, we discuss the Wells short exact sequence in the above context. [ABSTRACT FROM AUTHOR]

Recently, new combinatorial structures called disjoint partial difference families (DPDFs) and external partial difference families (EPDFs) were introduced, which simultaneously generalize partial difference sets, disjoint difference families and external difference families, and have applications in information security. So far, all known construction methods have used cyclotomy in finite fields. We present the first noncyclotomic infinite families of DPDFs which are also EPDFs, in structures other than finite fields (in particular cyclic groups and nonabelian groups). As well as direct constructions, we present an approach to constructing DPDFs/EPDFs using relative difference sets (RDSs); as part of this, we demonstrate how the well‐known RDS result of Bose extends to a very natural construction for DPDFs and EPDFs. [ABSTRACT FROM AUTHOR]

In this paper, we study parameter deformations of matrix valued orthogonal polynomials. These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail. [ABSTRACT FROM AUTHOR]

Let G be a non-abelian group and Z (G) be the center of G. The non-commuting graph Γ G associated to G is the graph whose vertex set is G ∖ Z (G) and two distinct elements x , y are adjacent if and only if x y ≠ y x. We prove that if G and H are non-abelian groups with isomorphic non-commuting graphs, such that G is nilpotent, then H is nilpotent, provided | Z (G) | ≥ | Z (H) |. Actually we prove conjecture 3 proposed in V. Grazian and C. Monetta (2023) [5]. [ABSTRACT FROM AUTHOR]

For any positive integer n, let C n be the cyclic group of order n. We determine all possible values of the integer group determinants for the non-abelian groups C 2 2 ⋊ C 4 , C 4 ⋊ C 4 and C 8 ⋊ 5 C 2 of order 16. [ABSTRACT FROM AUTHOR]

We ask the question of classical super (non-)Abelian T-duality for BTZ black hole metric coupling to two fermionic fields. Our approach is based on super Poisson-Lie (PL) T-duality in the presence of spectator fields. In order to study the Abelian T-duality of the metric we dualize over the Abelian Lie supergroups of the types (1|2) and (2|2), in such a way that it is shown that both original and dual backgrounds of the models are conformally invariant up to one-loop order in the presence of field strength. Then, we study the non-Abelian T-duality of the BTZ vacuum metric coupling to two fermionic fields. The dualizing is performed on some non-Abelian Lie supergroups of the type (2|2), in such a way that we are dealing with semi-Abelian superdoubles which are non-isomorphic as Lie superalgebras in each of the models. In the non-Abelian T-duality case, it is interesting to mention that the models can be conformally invariant up to one-loop order in both cases of the absence and presence of field strength. In addition, starting from the decomposition of semi-Abelian Drinfeld superdoubles generated by some of the C 3 ⨁ A 1,1 Lie superbialgebras we study the super PL T-plurality of the BTZ vacuum metric coupled to two fermionic fields. However, our findings are interesting in themselves, but at a constructive level, can prompt many new insights into supergravity and manifestly have interesting mathematical relationships with double field theory. [ABSTRACT FROM AUTHOR]

We study the joint distribution of random abelian and non-abelian groups. In the abelian case, we prove several universality results for the joint distribution of the multiple cokernels for random p -adic matrices. In the non-abelian case, we compute the joint distribution of random groups given by the quotients of the free profinite group by random relations. In both cases, we generalize the known results on the distribution of the cokernels of random p -adic matrices and random groups. Our proofs are based on the observation that mixed moments determine the joint distribution of random groups, which extends the works of Wood for abelian groups and Sawin for non-abelian groups. [ABSTRACT FROM AUTHOR]

By definition, a group is called narrow if it does not contain a copy of a non-abelian free group. We describe the structure of finite and narrow normal subgroups in Coxeter groups and their automorphism groups. [ABSTRACT FROM AUTHOR]

In this paper, we study nonabelian extensions of associative algebras using associative 2-algebra homomorphisms. First we construct an associative 2-algebra using the bimultipliers of an associative algebra. Then we classify nonabelian extensions of associative algebras using associative 2-algebra homomorphisms. Finally we analyze the relation between nonabelian extensions of associative algebras and nonabelian extensions of the corresponding commutator Lie algebras. [ABSTRACT FROM AUTHOR]

A topological descriptor is a numerical value derived from the molecular structure that encapsulates the most important structural characteristics of the molecule under consideration. Fundamentally, it involves assigning an algebraic value to the composition of chemicals while developing a relationship between this value and several physical properties, like biological activity, and chemical reactivity. This article examines multiple kinds of degree and eccentricity-based topological indices for power graphs of various finite groups. We calculate the Wiener index and its reciprocal, atom-bond connectivity index and its fourth version, the Schultz index, the geometric–arithmetic and harmonic indices, and finally determine the general Randić and Harary indices of power graphs of finite cyclic and non-cyclic groups of order p q , dihedral, and generalized quaternion groups, where p , q (q ≥ p) are distinct primes. [ABSTRACT FROM AUTHOR]

Local constraint in the lattice gauge theory provides an exotic mechanism that facilitates the disorder-free localization. However, the understanding of nonequilibrium dynamics in the non-Hermitian lattice gauge model remains limited. Here, we investigate the quench dynamics in a system of spinless fermions with nonreciprocal hopping in the Z 2 gauge field. By employing a duality mapping, we systematically explore the non-Hermitian skin effect, localization-delocalization transition, and real-complex transition. Through the identification of diverse scaling behaviors of quantum mutual information for fermions and spins, we propose that the non-Hermitian quantum disentangled liquids exist both in the localized and delocalized phases, the former originates from the Z 2 gauge field and the latter arises from the non-Hermitian skin effect. Furthermore, we demonstrate that the nonreciprocal dissipation causes the flow of quantum information. Our results provide valuable insights into the nonequilibrium dynamics in the gauge field, and may be experimentally validated using quantum simulators. Lattice gauge theory, a subset of gauge theory, has been successfully applied to a range of quantum systems allowing for the investigation of localised phenomena within these systems. Here, the authors consider a non-Hermitian lattice model observing a quantum disentangled liquid state that exists in both the localised and delocalised phases. [ABSTRACT FROM AUTHOR]

We prove that every finitely generated, residually finite group G$G$ embeds into a finitely generated perfect branch group Γ$\Gamma$ such that many properties of G$G$ are preserved under this embedding. Among those are the properties of being torsion, being amenable and not containing a non‐abelian free group. As an application, we construct a finitely generated, non‐amenable torsion branch group. [ABSTRACT FROM AUTHOR]

We consider a natural generalization of the concept of order of an element in a group G : an element g ∈ G is said to have order k in a subgroup H (respectively, in a coset H u) of G if k is the first strictly positive integer such that g k ∈ H (respectively, g k ∈ H u). We study this notion and its algorithmic properties in the realm of free groups and some related families. Both positive and negative (algorithmic) results emerge in this setting. On the positive side, among other results, we prove that the order of elements, the set of orders (called spectrum), and the set of preorders (i.e. the set of elements of a given order) with respect to finitely generated subgroups are always computable in free and free times free-abelian groups. On the negative side, we provide examples of groups and subgroups having essentially any subset of natural numbers as relative spectrum; in particular, non-recursive and even not computably enumerable sets of natural numbers. Also, we take advantage of Mikhailova's construction to see that the spectrum membership problem is unsolvable for direct products of nonabelian free groups. [ABSTRACT FROM AUTHOR]

We describe a general method for deducing T-dualities of little string theories, which are dualities between these theories that arise when they are compactified on circle. The method works for both untwisted and twisted circle compactifications of little string theories and is based on surface geometries associated to these circle compactifications. The surface geometries describe information about Calabi-Yau threefolds on which M-theory can be compactified to construct the corresponding circle compactified little string theories. Using this method, we deduce at least one T-dual, and in some cases multiple T-duals, for untwisted and twisted circle compactifications of most of the little string theories that can be described on their tensor branches in terms of a 6d supersymmetric gauge theory with a simple non-abelian gauge group, which are also known as rank-0 little string theories. This includes little string theories carrying N = (1, 1) and N = (1, 0) supersymmetries. For many, but not all, circle compactifications of N = (1, 1) little string theories, we have T-dualities that realize Langlands dualities between affine Lie algebras. Along the way, we find another discrete theta angle distinct from 0 and π for an E-string node. [ABSTRACT FROM AUTHOR]

A nontrivial knot is called minimal if its knot group does not surject onto the knot groups of other nontrivial knots. In this paper, we determine the minimality of the rational knots C (2 n + 1 , 2 m , 2) in the Conway notation, where m ≠ 0 and n ≠ 0 , − 1 are integers. When | m | ≥ 2 , we show that the nonabelian SL 2 (ℂ) -character variety of C (2 n + 1 , 2 m , 2) is irreducible and therefore C (2 n + 1 , 2 m , 2) is a minimal knot. The proof of this result is an interesting application of Eisenstein's irreducibility criterion for polynomials over integral domains. [ABSTRACT FROM AUTHOR]

Let G be a finite non-abelian group and let T be a transversal of the center of G in G. The non-commuting graph of G on a transversal of the center is the graph whose vertices are the non-central elements of T and two vertices x and y are joined by an edge whenever xy ≠ yx. In this paper, we classify the groups whose non-commuting graph on a transversal of the center is projective. [ABSTRACT FROM AUTHOR]

Let G be a finite non-abelian group. The commuting conjugacy class graph Γ ⁢ (G) is defined as a graph whose vertices are non-central conjugacy classes of G and two distinct vertices X and Y in Γ ⁢ (G) are connected by an edge if there exist elements x ∈ X and y ∈ Y such that x ⁢ y = y ⁢ x . In this paper, the structure of the commuting conjugacy class graph of group G with the property that G Z ⁢ (G) is isomorphic to a Frobenius group of order pq or p 2 ⁢ q , is determined. [ABSTRACT FROM AUTHOR]

The original article can be found online at https://doi.org/10.1007/s00222-024-01262-4The publication of this article unfortunately contained mistakes. The style of the references was not correct. The corrected reference list is given below.The original article has been corrected.Publisher's NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.By Will Sawin and Melanie Matchett WoodReported by Author; Author [Extracted from the article]

We prove that if L = F 4 2 ⁢ (2 2 ⁢ n + 1) ′ and 푥 is a nonidentity automorphism of 퐿, then G = ⟨ L , x ⟩ has four elements conjugate to 푥 that generate 퐺. This result is used to study the following conjecture about the 휋-radical of a finite group. Let 휋 be a proper subset of the set of all primes and let 푟 be the least prime not belonging to 휋. Set m = r if r = 2 or 3 and m = r − 1 if r ⩾ 5 . Supposedly, an element 푥 of a finite group 퐺 is contained in the 휋-radical O π ⁡ (G) if and only if every 푚 conjugates of 푥 generate a 휋-subgroup. Based on the results of this and previous papers, the conjecture is confirmed for all finite groups whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, unitary simple group, or to one of the groups of type B 2 2 ⁢ (2 2 ⁢ n + 1) , G 2 2 ⁢ (3 2 ⁢ n + 1) , F 4 2 ⁢ (2 2 ⁢ n + 1) ′ , G 2 ⁢ (q) , or D 4 3 ⁢ (q) . [ABSTRACT FROM AUTHOR]

We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the lawlessness growth function A Γ : N → N . We show that A Γ is bounded if and only if Γ has a non-abelian free subgroup. By contrast, we construct, for any non-decreasing unbounded function f : N → N , an elementary amenable lawless group for which A Γ grows more slowly than 푓. We produce torsion lawless groups for which A Γ is at least linear using Golod–Shafarevich theory and give some upper bounds on A Γ for Grigorchuk's group and Thompson's group 퐅. We note some connections between A Γ and quantitative versions of residual finiteness. Finally, we also describe a function M Γ quantifying the property of Γ having no mixed identities and give bounds for non-abelian free groups. By contrast with A Γ , there are no groups for which M Γ is bounded: we prove a universal lower bound on M Γ ⁢ (n) of the order of log ⁡ (n) . [ABSTRACT FROM AUTHOR]

Some constructions in string theory are linked to noncommutative geometry. Non-geometric strings are among such constructions. Non-geometric strings are related to a noncommutative torus. In this article, we discuss some aspects of non-geometric heterotic strings. We review the construction of eight-dimensional (8D) non-geometric heterotic strings, proposed by Malmendier and Morrison, which do not allow for a geometric interpretation. In the construction, the e 8 ⊕ e 7 gauge algebra is unbroken. The moduli space of 8D non-geometric heterotic strings and theories arising in the moduli space can be analyzed by studying the geometries of elliptically fibered K3 surfaces with a global section by applying F-theory/heterotic duality. In addition, we review the results of the points in the 8D non-geometric heterotic moduli with the unbroken e 8 ⊕ e 7 gauge algebra, at which the non-Abelian gauge groups are maximally enhanced. At these points, the gauge groups formed in the theories do not allow for a perturbative interpretation of the heterotic perspective. However, from the dual F-theory perspective, the K3 geometries at these points are deformations of the stable degenerations that arise from the coincident 7-branes. On the heterotic side, these enhancements can be understood as a non-perturbative effect of 5-brane insertions. [ABSTRACT FROM AUTHOR]

We give a pedagogical account of noncommutative gauge and gravity theories, where the exterior product between forms is deformed into a ⋆ -product via an abelian twist (e.g. the Groenewold–Moyal twist). The Seiberg–Witten map between commutative and noncommutative gauge theories is introduced. It allows to express the action of noncommutative Einstein gravity coupled to spinor fields in terms of the usual commutative action with commutative fields plus extra interaction terms dependent on the noncommutativity parameter. [ABSTRACT FROM AUTHOR]

We study two variants of the cosine subtraction law on a semigroup S. The main objective is to solve g (x y ∗) = g (x) g (y) + f (x) f (y) for unknown functions g , f : S → C , where x ↦ x ∗ is an anti-homomorphic involution. Until now this equation has not been solved on non-commutative semigroups, nor even on non-Abelian groups with x ∗ : = x - 1 . We solve this equation on semigroups under the assumption that g is central, and on groups generated by their squares under the assumption that x ∗ : = x - 1 . In addition we give a new proof for the solution of the variant g (x σ (y)) = g (x) g (y) + f (x) f (y) , where σ : S → S is a homomorphic involution. The continuous solutions on topological semigroups are also found. [ABSTRACT FROM AUTHOR]

The neutrino sector offers one of the most sensitive probes of new physics beyond the Standard Model of Particle Physics (SM). The mechanism of neutrino mass generation is still unknown. The observed suppression of neutrino masses hints at a large scale, conceivably of the order of the scale of a rand unified theory (GUT), which is a unique feature of neutrinos that is not shared by the charged fermions. The origin of neutrino masses and mixing is part of the outstanding puzzle of fermion masses and mixings, which is not explained ab initio in the SM. Flavor model building for both quark and lepton sectors is important in order to gain a better understanding of the origin of the structure of mass hierarchy and flavor mixing, which constitute the dominant fraction of the SM parameters. Recent activities in neutrino flavor model building based on non-Abelian discrete flavor symmetries and modular flavor symmetries have been shown to be a promising direction to explore. The emerging models provide a framework that has a significantly reduced number of undetermined parameters in the flavor sector. In addition, such a framework affords a novel origin of C P violation from group theory due to the intimate connection between physical C P transformation and group theoretical properties of non-Abelian discrete groups. Model building based on non-Abelian discrete flavor symmetries and their modular variants enables the particle physics community to interpret the current and anticipated upcoming data from neutrino experiments. Non-Abelian discrete flavor symmetries and their modular variants can result from compactification of a higher-dimensional theory. Pursuit of flavor model building based on such frameworks thus also provides the connection to possible UV completions: in particular, to string theory. We emphasize the importance of constructing models in which the uncertainties of theoretical predictions are smaller than, or at most compatible with, the error bars of measurements in neutrino experiments. While there exist proof-of-principle versions of bottom-up models in which the theoretical uncertainties are under control, it is remarkable that the key ingredients of such constructions were discovered first in top-down model building. We outline how a successful unification of bottom-up and top-down ideas and techniques may guide us towards a new era of precision flavor model building in which future experimental results can give us crucial insights into the UV completion of the SM. [ABSTRACT FROM AUTHOR]

We show that if a locally compact group |$G$| is non-abelian, then the amenability constant of its Fourier algebra is |$\geq 3/2$|⁠ , extending a result of [ 9 ] who proved that this holds for finite non-abelian groups. Our lower bound, which is known to be best possible, improves on results by previous authors and answers a question raised by [ 16 ]. To do this, we study a minorant for the amenability constant, related to the anti-diagonal in |$G\times G$|⁠ , which was implicitly used in Runde's work but hitherto not studied in depth. Our main novelty is an explicit formula for this minorant when |$G$| is a countable virtually abelian group, in terms of the Plancherel measure for |$G$|⁠. As further applications, we characterize those non-abelian groups where the minorant attains its minimal value and present some examples to support the conjecture that the minorant always coincides with the amenability constant. [ABSTRACT FROM AUTHOR]

Topological quantum error correction based on the manipulation of the anyonic defects constitutes one of the most promising frameworks towards realizing fault-tolerant quantum devices. Hence, it is crucial to understand how these defects interact with external defects such as boundaries or domain walls. Motivated by this line of thought, in this work, we study the fusion events between anyons in the bulk and at the boundary in fixed-point models of 2 + 1-dimensional non-chiral topological order defined by arbitrary fusion categories. Our construction uses generalized tube algebra techniques to construct a bi-representation of bulk and boundary defects. We explicitly derive a formula to calculate the fusion multiplicities of a bulk-to-boundary fusion event for twisted quantum double models and calculate some exemplary fusion events for Abelian models and the (twisted) quantum double model of S3, the simplest non-Abelian group-theoretical model. Moreover, we use the folding trick to study the anyonic behavior at non-trivial domain walls between twisted S3 and twisted Z 2 as well as Z 3 models. A recurring theme in our construction is an isomorphism relating twisted cohomology groups to untwisted ones. The results of this work can directly be applied to study logical operators in two-dimensional topological error correcting codes with boundaries described by a twisted gauge theory of a finite group. [ABSTRACT FROM AUTHOR]

Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and unify these results in two different ways. Firstly, we work over arbitrary non-abelian finite groups admitting an abelian subgroup of index 2. Secondly, our main result actually characterizes integral mixed Cayley graphs over such finite groups, in the spirit of a very recent result of Kadyan–Bhattarcharjya in the abelian case. [ABSTRACT FROM AUTHOR]

We determine the finite non‐abelian simple groups which occur as the type of a Hopf–Galois structure on a solvable extension. In the language of skew braces, our result gives a complete list of finite non‐abelian simple groups which occur as the additive group of a skew brace with solvable multiplicative group. [ABSTRACT FROM AUTHOR]

Multi-Higgs models equipped with global symmetry groups, either exact or softly broken, offer a rich framework for constructions beyond the Standard Model and lead to remarkable phenomenological consequences. Knowing all the symmetry options within each class of models can guide its phenomenological exploration, as confirmed by the vast literature on the two- and three-Higgs-doublet models. Here, we begin a systematic study of finite non-abelian symmetry groups which can be imposed on the scalar sector of the four-Higgs-doublet model (4HDM) without leading to accidental symmetries. In this work, we derive the full list of such non-abelian groups available in the 4HDM that can be constructed as extensions of cyclic groups by their automorphism groups. This list is remarkably restricted but it contains cases which have not been previously studied. Since the methods we develop may prove useful for other classes of models, we present them in a pedagogical manner. [ABSTRACT FROM AUTHOR]

Torsion in the concordance group of knots in S 3 can be studied with the algebraic concordance group . Here, is a field of characteristic χ () ≠ 2. The group was defined by Levine, who also obtained an algebraic classification when = ℚ. While the concordance group is abelian, it embeds into the non-abelian virtual knot concordance group . It is unknown if admits non-classical finite torsion. Here, we define the virtual algebraic concordance group for Seifert surfaces of almost classical knots. This is an analogue of for homologically trivial knots in thickened surfaces Σ × [ 0 , 1 ] , where Σ is closed and oriented. The main result is an algebraic classification of . A consequence of the classification is that ℚ embeds into ℚ and ℚ contains many nontrivial finite-order elements that are not algebraically concordant to any classical Seifert matrix. For = ℤ / 2 ℤ , we give a generalization of the Arf invariant. [ABSTRACT FROM AUTHOR]

We construct finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite-dimensional pointed Hopf algebra over a nonabelian group with nonsimple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch-Schneider. Our starting point is the classification of finite-dimensional Nichols algebras over nonabelian groups by Heckenberger-Vendramin, which consist of low-rank exceptions and large-rank families. We prove that the large-rank families are cocycle twists of Nichols algebras constructed by the second author as foldings of Nichols algebras of Cartan type over abelian groups by outer automorphisms. This enables us to give uniform Lie-theoretic descriptions of the large-rank families, prove generation in degree 1, and construct liftings. We also show that every lifting is a cocycle deformation of the corresponding coradically graded Hopf algebra using an explicit presentation by generators and relations of the Nichols algebra. On the level of tensor categories, we construct families of graded extensions of the representation category of a quantum group by a group of diagram automorphism. [ABSTRACT FROM AUTHOR]

Let K be a local field with perfect residue field and let L/K be a finite Galois extension. The Hasse-Arf theorem says that if \operatorname {Gal}(L/K) is abelian then the upper ramification breaks of L/K must be integers. We prove the following converse to the Hasse-Arf theorem: Let G be a nonabelian group which is isomorphic to the Galois group of some totally ramified extension E/F of local fields with residue characteristic p>2. Then there is a totally ramified extension L/K of local fields with residue characteristic p such that \operatorname {Gal}(L/K)\cong G and L/K has at least one nonintegral upper ramification break. [ABSTRACT FROM AUTHOR]

Onsite gain-loss-induced topological braiding principle of non-Hermitian energy bands is theoretically formulated in multiband lattice models with Hermitian hopping amplitudes. Braid phase transition occurs when the gain-loss parameter is tuned across exceptional point degeneracy. Laboratory realizable effective-Hamiltonians are proposed to realize braid groups B 2 and B 3 of two and three bands, respectively. While B 2 is trivially Abelian, the group B 3 features non-Abelian braiding and energy permutation originating from the collective behavior of multiple exceptional points. Phase diagrams with respect to lattice parameters to realize braid group generators and their non-commutativity are shown. The proposed theory is conducive to synthesizing exceptional materials for applications in topological computation and information processing. [ABSTRACT FROM AUTHOR]

A graph is edge-primitive if its automorphism group acts primitively on the edge set of the graph. Edge-primitive graphs form an important subclass of symmetric graphs. In this paper, edge-primitive graphs of order as a product of two distinct primes are completely determined. This depends on non-abelian simple groups with a subgroup of index pq being classified, where p > q are odd primes. [ABSTRACT FROM AUTHOR]

We consider cyclic branched coverings of a 3-parameter family of rational knots in S 3 and study the left orderability of their fundamental groups. We first compute the nonabelian SL 2 (ℂ) -character varieties of the rational knots C (2 p , 2 m , 2 n + 1) in the Conway notation, where p , m , n are integers. We then study real points on these varieties and finally use them to determine the left orderability of the fundamental groups of cyclic branched coverings of C (2 p , 2 m , 2 n + 1). [ABSTRACT FROM AUTHOR]

A subgroup of a group is -permutable in if for every subgroup there exists such that . A subgroup is hereditarily -permutable in if is -permutable in every subgroup of which includes . The Kourovka Notebook has Problem 17.112(b): Which finite nonabelian simple groups possess a proper hereditarily -permutable subgroup? We answer this question for the exceptional groups of Lie type. Moreover, for the Suzuki groups we prove that a proper subgroup of is -permutable if and only if the order of the subgroup is 2. In particular, we obtain an infinite series of groups with -permutable subgroups. [ABSTRACT FROM AUTHOR]

The factor graph of an instance of a constraint satisfaction problem with n variables and m constraints is the bipartite graph between [m] and [n] describing which variable appears in which constraints. Thus, an instance of a CSP is completely determined by its factor graph and the list of predicates. We show optimal inapproximability of Max-3-LIN over non-Abelian groups (both in the perfect completeness case and in the imperfect completeness case), even when the factor graph is fixed. Previous reductions which proved similar optimal inapproximability results produced factor graphs that were dependent on the input instance. Along the way, we also show that these optimal hardness results hold even when we restrict the linear equations in the Max-3-LIN instances to the form x · y · z = g , where x, y, z are the variables and g is a group element. We use representation theory and Fourier analysis over non-Abelian groups to analyze the reductions. [ABSTRACT FROM AUTHOR]