Finite Nonabelian Group Research Articles

Finite p-groups with Few Kernels of Nonlinear Irreducible Characters

Let G be a finite nonabelian group and Kern(G) the set of kernels of nonlinear irreducible characters of G. In this paper, finite p-groups G with ∣Kern(G)∣ ≤ 3 are determined.

Minimal Euler characteristics for even-dimensional manifolds with finite fundamental group

Abstract We consider the Euler characteristics $\chi (M)$ of closed, orientable, topological $2n$ -manifolds with $(n-1)$ -connected universal cover and a given fundamental group G of type $F_n$ . We define $q_{2n}(G)$ , a generalised version of the Hausmann-Weinberger invariant [19] for 4–manifolds, as the minimal value of $(-1)^n\chi (M)$ . For all $n\geq 2$ , we establish a strengthened and extended version of their estimates, in terms of explicit cohomological invariants of G. As an application, we obtain new restrictions for nonabelian finite groups arising as fundamental groups of rational homology 4–spheres.

On the spectra of non-commuting graphs of certain class of groups

Let [Formula: see text] be the center of a finite non-abelian group [Formula: see text] The non-commuting graph of [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] A group [Formula: see text] is called a CA-group if the centralizer [Formula: see text] for every non central element [Formula: see text] is abelian. In this paper, we investigate the distance, distance (signless) Laplacian spectra of non-commuting graphs of some CA-groups, and obtain some conditions on them so that the corresponding non-commuting graph is distance, distance (signless) Laplacian integral.

Groups with maximum vertex degree commuting graphs

Let G be a finite non-abelian group and Z(G) be its center. We associate a commuting graph \(\Gamma (G)\) to G, whose vertex set is \(G\setminus Z(G)\) and two distinct vertices are adjacent if they commute. In this paper we prove that the set of all non-abelian groups whose commuting graph has maximum vertex degree bounded above by a fixed \(k \in {\mathbb {N}}\) is finite. Also, we characterize all groups for which the associated commuting graphs have maximum vertex degree at most 4.

C−P−T fractionalization

Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as $\mathbb{Z}_2$-involutions in the passive transformation on the spacetime coordinates; but together with a charge conjugation C, the total C-P-R-T symmetries have enriched active transformations on fields in representations of the spacetime-internal symmetry groups of quantum field theories (QFTs). In this work, we derive that these symmetries can be further fractionalized, especially in the presence of the fermion parity $(-1)^{\rm F}$. We elaborate on examples including relativistic Lorentz invariant QFTs (e.g., spin-1/2 Dirac or Majorana spinor fermion theories) and nonrelativistic quantum many-body systems (involving Majorana zero modes), and comment on applications to spin-1 Maxwell electromagnetism (QED) or interacting Yang-Mills (QCD) gauge theories. We discover various C-P-R-T-$(-1)^{\rm F}$ group structures, e.g., Dirac spinor is in a projective representation of $\mathbb{Z}_2^{\rm C}\times \mathbb{Z}_2^{\rm P} \times \mathbb{Z}_2^{\rm T}$ but in an (anti)linear representation of an order-16 nonabelian finite group, as the central product between an order-8 dihedral (generated by C and P) or quaternion group and an order-4 group generated by T with T$^2=(-1)^{\rm F}$. The general theme may be coined as C-P-T or C-R-T fractionalization.

Counting Centralizers of a Finite Group with an Application in Constructing the Commuting Conjugacy Class Graph

The set of all centralizers of elements in a finite group G is denoted by Cent(G) and G is called n-centralizer if . In this paper, the structure of centralizers in a non-abelian finite group G with this property that is obtained. As a consequence, it is proved that such a group has exactly element centralizers and the structure of the commuting conjugacy class graph of G is completely determined.

Anomalies of non-Abelian finite groups via cobordism

We use cobordism theory to analyse anomalies of finite non-abelian symmetries in 4 spacetime dimensions. By applying the method of ‘anomaly interplay’, which uses functoriality of cobordism and naturality of the η-invariant to relate anomalies in a group of interest to anomalies in other (finite or compact Lie) groups, we derive the anomaly for every representation in many examples motivated by flavour physics, including S3, A4, Q8, and SL(2, \U0001d53d3).In the case of finite abelian groups, it is well known that anomalies can be ‘truncated’ in a way that has no effect on low-energy physics, by means of a group extension. We extend this idea to non-abelian symmetries. We show, for example, that a system with A4 symmetry can be rendered anomaly-free, with only one-third as many fermions as naïvely required, by passing to a larger symmetry. As another example, we find that a well-known model of quark and lepton masses utilising the SL(2, \U0001d53d3) symmetry is anomalous, but that the anomaly can be cancelled by enlarging the symmetry to a ℤ/3 extension of SL(2, \U0001d53d3).

Certain Topological Indices of Non-Commuting Graphs for Finite Non-Abelian Groups.

A topological index is a number derived from a molecular structure (i.e., a graph) that represents the fundamental structural characteristics of a suggested molecule. Various topological indices, including the atom-bond connectivity index, the geometric–arithmetic index, and the Randić index, can be utilized to determine various characteristics, such as physicochemical activity, chemical activity, and thermodynamic properties. Meanwhile, the non-commuting graph of a finite group is a graph where non-central elements of are its vertex set, while two different elements are edge connected when they do not commute in . In this article, we investigate several topological properties of non-commuting graphs of finite groups, such as the Harary index, the harmonic index, the Randić index, reciprocal Wiener index, atomic-bond connectivity index, and the geometric–arithmetic index. In addition, we analyze the Hosoya characteristics, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of the non-commuting graphs over finite subgroups of . We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups.

Abelian maps, brace blocks, and solutions to the Yang-Baxter equation

Let G be a finite nonabelian group. We show how an endomorphism of G with abelian image gives rise to a family of binary operations { ∘ n : n ∈ Z ≥ 0 } on G such that ( G , ∘ m , ∘ n ) is a skew left brace for all m , n ≥ 0 . A brace block gives rise to a number of non-degenerate set-theoretic solutions to the Yang-Baxter equation. We give examples showing that the number of solutions obtained can be arbitrarily large.

A Brief Survey on the Inverse Galois Problem

Inverse Galois problem (IGP) states whether any finite group is realizable as a Galois group over the field K. It is the question of the structure and representation of the Galois group and also questions its epimorphic images. So, it is called an inverse Galois problem. For K=ℚ (the field of rational number), it is called a classical inverse Galois problem (CIGP). This paper reviews the positive answer to the classical inverse Galois problem (CIGP) for all finite abelian groups and some finite non-abelian solvable groups. We also discuss this problem (CIGP) for some finite non-solvable groups in this paper. This problem still remains to solve, but if we find the true value of the statement ‘All subgroups of order m of the symmetric group (Sm) for all m are realizable as Galois group over ℚ’ then its truth value gives the answer of CIGP. We check this statement for m=1,2,3,4 and 5 in this paper, where we get that this statement is true. If this statement is true, then CIGP has a positive answer. But if this statement is false then CIGP has a negative answer.

On the Topological Indices of Commuting Graphs for Finite Non-Abelian Groups

A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It is possible to determine several different properties, such as chemical activity, thermodynamic properties, physicochemical activity, and biological activity, using several topological indices, such as the geometric-arithmetic index, arithmetic-geometric index, Randić index, and the atom-bond connectivity indices. Consider G as a group and H as a non-empty subset of G. The commuting graph C(G,H), has H as the vertex set, where h1,h2∈H are edge connected whenever h1 and h2 commute in G. This article examines the topological characteristics of commuting graphs having an algebraic structure by computing their atomic-bond connectivity index, the Wiener index and its reciprocal, the harmonic index, geometric-arithmetic index, Randić index, Harary index, and the Schultz molecular topological index. Moreover, we study the Hosoya properties, such as the Hosoya polynomial and the reciprocal statuses of the Hosoya polynomial of the commuting graphs of finite subgroups of SL(2,C). Finally, we compute the Z-index of the commuting graphs of the binary dihedral groups.

Connectivity of superpower graphs of some non-abelian finite groups

For a finite group [Formula: see text] the superpower graph [Formula: see text] of [Formula: see text] is an undirected simple graph with vertex set [Formula: see text] and two vertices in [Formula: see text] are adjacent in [Formula: see text] if and only if the order of one divides the order of the other in the group [Formula: see text]. In this paper, we give sharp bounds for the vertex connectivity of superpower graphs [Formula: see text] and [Formula: see text] of dihedral group [Formula: see text] and dicyclic group [Formula: see text]

The character degree ratio and large abelian subgroups

Let b(G) be the largest degree of the irreducible characters of a nonabelian finite group G and let m(G) be the smallest degree of the nonlinear irreducible characters of G. M. Isaacs defined the character degree ratio of G to be rat(G)=b(G)/m(G). He proved that for nonabelian solvable groups the derived length is bounded by a logarithmic function of the character degree ratio. In a similar spirit, results that restrict the structure of nonsolvable groups in terms of the character degree ratio were later proved by J. P. Cossey, M. Lewis and H. N. Nguyen. In this note we show that, with some rather precisely determined exceptions, G has an abelian subgroup of index bounded by a polynomial function in rat(G). As a consequence we get stronger (in most situations) versions of most of the previously known results on the character degree ratio.

Lattice model constructions for gapless domain walls between topological phases

Domain walls between different topological phases are one of the most interesting phenomena that reveal the non-trivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group $G$. As simple examples, we numerically study several finite groups(including both Abelian and non-Abelian finite group such as $S_3$) in $2$D using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.

Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras

Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.

Applications of group theory in crystallography

Group theory is a powerful tool for studying symmetric physical systems. Such systems include, in particular, molecules and crystals with symmetry. Group theory serves to explain the most important characteristics of atomic spectra. Group theory is also applied to the problems of atomic and nuclear physics. This paper gives examples of the use of the apparatus of group theory in research on crystallography, quantum mechanics, elementary particle physics. In particular, in these studies matrix groups and representations of unitary groups are actively used. For such groups we give an overview of the results on their recognition by the spectrum (by the orders of the elements of the group). This direction has been intensively developed in recent years both in our country and abroad. Recognition of finite simple non-Abelian groups by spectrum has been studied for last thirty years in Yekaterinburg at the Institute of Mathematics and Mechanics of the Ural Division of the Russian Academy of Sciences, in Chelyabinsk Federal University and in the Novosibirsk Institute of Mathematics of Siberian Division of the Russian Academy of Sciences. Some simple non-Abelian groups are not recognizable by their spectra. We have proposed an approach for recognizing groups by the bottom layer. The bottom layer of a group is the set of its elements of prime orders. A group is called recognizable by the bottom layer under additional conditions if it is uniquely restored by the bottom layer under these conditions. The paper considers some examples of simple non-Abelian finite groups that are not recognizable by spectra. For these examples, simultaneous recognition by spectrum and by the bottom layer is proved.

Analyzing on My Turf Game Over Some Finite Non-Abelian Groups

The aim of this paper is to solve the “ON MY TURF" game over some finite nonabelian groups. Also, it presents the following results: 1-) If G has odd order, and the set F contains the identity element, then the first player A has a winning strategy. If F does not contain the identity, then B has a winning strategy. 2-) If G has an even order with only one element of order two, there is a winning strategy related to set F. 3-) If G has an even order with only three elements of order two which generate a subgroup isomorphic to Z2 × Z2, there is a winning strategy related to the set F.

Double-toroidal and 1-planar non-commuting graph of a group

Let G be a finite non-abelian group and denote by Z(G) its center. The non-commuting graph of G on a transversal of the center is the graph whose vertices are the non-central elements of a transversal of Z(G) in G and two vertices x and y are adjacent whenever xy=yx. In this work, we classify the finite non-abelian groups whose non-commuting graph on a transversal of the centeris double-toroidal or 1-planar.

On g-Noncommuting Graph of a Finite Group Relative to Its Subgroups

Let H be a subgroup of a finite non-abelian group G and g∈G. Let Z(H,G)={x∈H:xy=yx,∀y∈G}. We introduce the graph ΔH,Gg whose vertex set is G\Z(H,G) and two distinct vertices x and y are adjacent if x∈H or y∈H and [x,y]≠g,g−1, where [x,y]=x−1y−1xy. In this paper, we determine whether ΔH,Gg is a tree among other results. We also discuss about its diameter and connectivity with special attention to the dihedral groups.

Proportions of vanishing elements in finite groups

In this paper, we study the proportion of vanishing elements of finite groups. We show that the proportion of vanishing elements of every finite non-abelian group is bounded below by 1/2 and classify all finite groups whose proportions of vanishing elements attain this bound. For symmetric groups of degree at least 5, we show that this bound is at least 2327/2520 which is best possible.