Non-abelian Group Research Articles

On Edge-Primitive Graphs of Order as a Product of Two Distinct Primes

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.

Intersection configurations in free and free times free-abelian groups

In this paper, we study intersection configurations – which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability – in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group $G$ if there exist subgroups $H_1,\ldots, H_k \leqslant G$ realizing it. It is well known that free groups ${\mathbb {F}_{n}}$ satisfy the Howson property: the intersection of any two finitely generated subgroups is again finitely generated. We show that the Howson property is indeed the only obstruction for multiple intersection configurations to be realizable within nonabelian free groups. On the contrary, FTFA groups ${\mathbb {F}_{n}} \times \mathbb {Z}^m$ are well known to be non-Howson. We also study multiple intersections within FTFA groups, providing an algorithm to decide, given $k\geq 2$ finitely generated subgroups, whether their intersection is again finitely generated and, in the affirmative case, compute a ‘basis’ for it. We finally prove that any intersection configuration is realizable in an FTFA group ${\mathbb {F}_{n}} \times \mathbb {Z}^m$, for $n\geq 2$ and large enough $m$. As a consequence, we exhibit finitely presented groups where every intersection configuration is realizable.

Pointed Hopf algebras over nonabelian groups with nonsimple standard braidings

AbstractWe 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.

Several Zagreb indices of power graphs of finite non-abelian groups

Molecular topology can be described by using topological indices. These are quantitative measures of the essential structural features of a proposed molecule calculated from its molecular structure. It is a numerical value obtained from a molecular configuration that reflects the significant physical characteristics of the suggested molecule. Numerous physical properties, chemical reactivity, and biological activity are correlated with the chemical composition using an algebraic number. The power graph P(G) of a finite group G is a graph whose vertex set is G and in which two distinct vertices are connected by an edge when one element is an integral power of the other. This article investigates a wide range of degree-based topological descriptors for power graphs of various finite groups. We find numerous Zagreb indices (given in Table 1) of power graphs of finite non-cyclic and cyclic groups, dihedral, and generalized quaternion groups.

Constructing uniform 2-factorizations via row-sum matrices: Solutions to the Hamilton-Waterloo problem

In this paper, we formally introduce the concept of a row-sum matrix over an arbitrary group G. When G is cyclic, these types of matrices have been widely used to build uniform 2-factorizations of small Cayley graphs (or, Cayley subgraphs of blown-up cycles), which themselves factorize complete (equipartite) graphs.Here, we construct row-sum matrices over a class of non-abelian groups, the generalized dihedral groups, and we use them to construct uniform 2-factorizations that solve infinitely many open cases of the Hamilton-Waterloo problem, thus filling up large parts of the gaps in the spectrum of orders for which such factorizations are known to exist.

The Tits alternative for two-dimensional Artin groups and Wise's power alternative

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 an and bn either commute or generate a non-abelian free subgroup.

Dihedral lattice gauge theories on a quantum annealer

We study lattice gauge theory with discrete, non-Abelian gauge groups. We extend the formalism of previous studies on D-Wave’s quantum annealer as a computing platform to finite, simply reducible gauge groups. As an example, we use the dihedral group D_{n} with n=3,4 on a two plaquette ladder for which we provide proof-of-principle calculations of the ground-state and employ the known time evolution formalism with Feynman clock states.

Tadpoles and gauge symmetries

The tadpole conjecture proposes that complex structure moduli stabilisation by fluxes that have low tadpole charge can be realised only at special points in moduli space, leading generically to (large) gauge symmetries. Here we provide an exhaustive survey of the gauge symmetries arising in F-theory flux compactifications on products of attractive K3 surfaces, with complex structure moduli fully stabilised. We compute the minimal rank of the left-over non-abelian gauge group for all flux configurations within the tadpole bound, finding that it is always non-zero. It decreases in a roughly linear fashion with the tadpole charge, reaching zero at charge 30. By working out possible gauge algebras for different values of the tadpole, we find that all simple ADE Lie algebras of rank ≤ 18 appear.

Enhanced power graphs of certain non-abelian groups

The enhanced power graph of a group [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] 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.

Spinning switches on a wreath product

We classify an algebraic phenomenon on several families of wreath products that can be seen as coming from a generalization of a puzzle about switches on the corners of a spinning table. Such puzzles have been written about and generalized since they were first popularized by Martin Gardner in 1979. In this paper, we build upon a paper of Bar Yehuda, Etzion, and Moran, a paper of Ehrenborg and Skinner, and a paper of Rabinovich to provide perhaps the fullest generalization yet, modeling both the switches and the spinning table as arbitrary finite groups combined via a wreath product. We classify large families of wreath products depending on whether or not they correspond to a solvable puzzle, completely classifying the puzzle in the case when the switches behave like abelian groups, constructing winning strategies for all wreath products that are p-groups, and providing novel examples for other puzzles where the switches behave like nonabelian groups, including the puzzle consisting of two interchangeable copies of the monster group M. Lastly, we provide a number of open questions and conjectures, and provide other suggestions of how to generalize some of these ideas further.

Hypermaps Over Non-Abelian Simple Groups and Strongly Symmetric Generating Sets

A generating pair $x, y$ for a group $G$ is said to be symmetric if there exists an automorphism $\varphi_{x,y}$ of $G$ inverting both $x$ and $y$, that is, $x^{\varphi_{x,y}}=x^{-1}$ and $y^{\varphi_{x,y}}=y^{-1}$. Similarly, a group $G$ is said to be strongly symmetric if $G$ can be generated with two elements and if all generating pairs of $G$ are symmetric.
 In this paper we classify the finite strongly symmetric non-abelian simple groups. Combinatorially, these are the finite non-abelian simple groups $G$ such that every orientably regular hypermap with monodromy group $G$ is reflexible.

Quantifying lawlessness in finitely generated groups

Abstract We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the lawlessness growth function A Γ : N → N \mathcal{A}_{\Gamma}\colon\mathbb{N}\to\mathbb{N} . We show that A Γ \mathcal{A}_{\Gamma} 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 f\colon\mathbb{N}\to\mathbb{N} , an elementary amenable lawless group for which A Γ \mathcal{A}_{\Gamma} grows more slowly than 𝑓. We produce torsion lawless groups for which A Γ \mathcal{A}_{\Gamma} is at least linear using Golod–Shafarevich theory and give some upper bounds on A Γ \mathcal{A}_{\Gamma} for Grigorchuk’s group and Thompson’s group 𝐅. We note some connections between A Γ \mathcal{A}_{\Gamma} and quantitative versions of residual finiteness. Finally, we also describe a function M Γ \mathcal{M}_{\Gamma} quantifying the property of Γ having no mixed identities and give bounds for non-abelian free groups. By contrast with A Γ \mathcal{A}_{\Gamma} , there are no groups for which M Γ \mathcal{M}_{\Gamma} is bounded: we prove a universal lower bound on M Γ ⁢ ( n ) \mathcal{M}_{\Gamma}(n) of the order of log ⁡ ( n ) \log(n) .

Obstruction theories of extensions via track categories

Abelian track categories can be classified via the third Baues–Wirsching cohomology of small categories. This approach is used in this paper to derive the different generalizations of the obstruction theory of non-abelian group extensions, due to Cegarra, Garzón and Grandjean, Cegarra, Garzón and Ortega, and Chen, Du and Wang as particular cases of a general theory of abelian track categories and to give unified proofs of the main properties of these obstruction classes.

(p − 1)-bracket for Dp-branes in large R-R field background

The volume-preserving diffeomorphism is a key feature that characterizes the large constant R-R (p − 1)-form field background in a Dp-brane theory. It represents a symmetry of the theory that preserves the volume of space. To describe this symmetry, we introduce the concept of the (p − 1)-bracket, which generates the volume-preserving diffeomorphism. The (p − 1)-bracket is a mathematical operation that acts on (p − 1)-forms and encodes the transformation of the background field under the symmetry. To generalize the (p − 1)-bracket, we can apply it to the non-Abelian one-form gauge field, which is relevant in gauge theories with non-Abelian gauge groups. This allows us to extend the concept of volume-preserving diffeomorphism and its associated symmetry to non-Abelian gauge theories. When considering D-branes and T-duality, we introduce the transverse coordinates of the branes. T-duality is a symmetry transformation that relates String Theory compactified on different backgrounds. It exchanges the momentum and winding modes of strings and leads to an equivalence between theories with different numbers of dimensions. By incorporating T-duality and the generalized bracket, a general expression for the action in Dp-branes can be derived when p ≤ 6. This result connects the existing construction of Dp-branes with our generalized bracket, illustrating the relationship between the symmetry and its associated transformations and the dynamics of the branes. In addition, we can discuss the non-Abelianization of the (p − 2)-form gauge potential. This process involves generalizing the concept of non-Abelian gauge fields to higher-form gauge potentials. By extending the Lagrangian description of a single D-brane to multiple D-branes, a similar Lagrangian description can be established for both cases, highlighting the common underlying structure and symmetry properties. Our developments demonstrate the interplay between symmetries, gauge fields, and D-brane dynamics, providing a deeper understanding of the underlying principles within D-branes.

On integral mixed Cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2

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.

An Explicit Minorant for the Amenability Constant of the Fourier Algebra

Abstract 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.

On the bottom layer in groups

The question of the possibility of restoring information about a group by its lower layer, that is, by the set of its elements of prime orders, is considered. The question is classical for mathematical modeling: restoration of the missing information about the object from the part of the preserved data. A group is said to be recognizable from the bottom layer under additional conditions if it is uniquely reconstructed from the bottom layer under these conditions. A group G is said to be almost recognizable from the bottom layer under additional conditions if there exists a finite number of pairwise nonisomorphic groups satisfying these conditions, with the same bottom layer as the group G. A group G is called unrecognizable from the bottom layer under additional conditions if there is an infinite the number of pairwise non-isomorphic groups that satisfy these conditions and have the same bottom layer as the group G. Results are given on the recognition of groups by the bottom layer in various classes of groups. The concept of recognizability by the lower layer was introduced by analogy with the actively studied recognizability by spectrum, that is, by the set of orders of group elements. In this paper, we consider groups that, without a single element, coincide with their bottom layer. Examples of groups with these conditions in the classes of Abelian and non-Abelian groups are given. The properties of such groups are established. The results can be applied when encoding information in space communications.

Hamiltonians and gauge-invariant Hilbert space for lattice Yang-Mills-like theories with finite gauge group

Motivated by quantum simulation, we consider lattice Hamiltonians for Yang-Mills gauge theories with finite gauge group, for example a finite subgroup of a compact Lie group. We show that the electric Hamiltonian admits an interpretation as a certain natural, nonunique Laplacian operator on the finite Abelian or non-Abelian group and derive some consequences from this fact. Independent of the chosen Hamiltonian, we provide a full explicit description of the physical, gauge-invariant Hilbert space for pure gauge theories and derive a simple formula to compute its dimension. We illustrate the use of the gauge-invariant basis to diagonalize a dihedral gauge theory on a small periodic lattice.

Construction of a three-parameter encryption scheme on Hermitian groups in the MST3 cryptosystem

The article proposes a method for constructing a three-parameter encryption scheme based on Hermitian groups, which improves the security parameters of the existing MST3 cryptosystem. The challenge of improving existing approaches to building cryptosystems is driven by successes in building a quantum computer with sufficient computing power to render many public-key cryptosystems insecure. In particular, we are talking about those cryptosystems based on the complexity of factorization or the discrete logarithm problem, such as RSA, ECC, etc. There are several proposals that have become classic over the past almost 20 years for using non-commutative groups to build quantum-resistant cryptosystems The unsolvable word problem is an interesting area of research for cryptosystem construction. It was formulated by Wagner and Magyarik and lies in the plane of application of permutation groups. Logarithmic signatures (LS) were proposed by Magliveras. In this context, the logarithmic signature is a special type of factorization, it is applied to finite groups. The latest version of this implementation is known as MST3 and is based on the Suzuki group.
 In 2008, Magliveras demonstrated a transitive limit of LS for the MST3 cryptosystem. Svaba later proposed the eMST3 cryptosystem with improved security options. A secret homomorphic cover was added for this improvement. Then, in 2018, T. van Trung proposed an MST3 approach using strong aperiodic LS for abelian p-groups. Kong and colleagues conducted an extensive analysis of MST3 and noted that since there are no publications yet on the quantum vulnerability of the algorithm, it can be considered a candidate for the post-quantum era.
 One valuable idea is to improve encryption efficiency by optimizing the computational overhead. This is done while reducing the large size of the key space. This approach can be applied to LS calculations outside the center of the group. And this was done over the final fields of the small dimensions using groups with high order.

Local types of (Γ,G)$(\Gamma ,G)$‐bundles and parahoric group schemes

AbstractLet be a simple algebraic group over an algebraically closed field . Let be a finite group acting on . We classify and compute the local types of ‐bundles on a smooth projective ‐curve in terms of the first nonabelian group cohomology of the stabilizer groups at the tamely ramified points with coefficients in . When , we prove that any generically simply connected parahoric Bruhat–Tits group scheme can arise from a ‐bundle. We also prove a local version of this theorem, that is, parahoric group schemes over the formal disc arise from constant group schemes via tamely ramified coverings.