Finite Subgroup Research Articles

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.

Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum \(E_n\), whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that \(E_n\) is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group \({\mathbb {G}}_n\) of the formal group in question. In this paper we find that the \({\mathbb {G}}_n\)-equivariant dual of \(E_n\) is in fact \(E_n\) twisted by a sphere with a non-trivial (when \(n>1\)) action by \({\mathbb {G}}_n\). This sphere is a dualizing module for the group \({\mathbb {G}}_n\), and we construct and study such an object \(I_{{\mathcal {G}}}\) for any compact p-adic analytic group \({\mathcal {G}}\). If we restrict the action of \({\mathcal {G}}\) on \(I_{{\mathcal {G}}}\) to certain type of small subgroups, we identify \(I_{{\mathcal {G}}}\) with a specific representation sphere coming from the Lie algebra of \({\mathcal {G}}\). This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of \(E_n^{hH}\) for select choices of p and n and finite subgroups H of \({\mathbb {G}}_n\).

Auslander’s theorem for dihedral actions on preprojective algebras of type A

AbstractGiven an algebra R and G a finite group of automorphisms of R, there is a natural map $\eta _{R,G}:R\#G \to \mathrm {End}_{R^G} R$ , called the Auslander map. A theorem of Auslander shows that $\eta _{R,G}$ is an isomorphism when $R=\mathbb {C}[V]$ and G is a finite group acting linearly and without reflections on the finite-dimensional vector space V. The work of Mori–Ueyama and Bao–He–Zhang has encouraged the study of this theorem in the context of Artin–Schelter regular algebras. We initiate a study of Auslander’s result in the setting of nonconnected graded Calabi–Yau algebras. When R is a preprojective algebra of type A and G is a finite subgroup of $D_n$ acting on R by automorphism, our main result shows that $\eta _{R,G}$ is an isomorphism if and only if G does not contain all of the reflections through a vertex.

Polynomials over ring of integers of global fields that have roots modulo every finite indexed subgroup

A polynomial with coefficients in the ring of integers OK of a global field K is called intersective if it has a root modulo every finite-indexed subgroup of OK. We prove two criteria for a polynomial f(x)∈OK[x] to be intersective. One of these criteria is in terms of the Galois group of the splitting field of the polynomial, whereas the second criterion is verifiable entirely in terms of constants which depend upon K and the polynomial f. The proofs use the theory of global field extensions and upper bound on the least prime ideal in the Chebotarev density theorem.

Alternating sign matrices of finite multiplicative order

We investigate alternating sign matrices that are not permutation matrices, but have finite order in a general linear group. We classify all such examples of the form P+T, where P is a permutation matrix and T has four non-zero entries, forming a square with entries 1 and −1 in each row and column. We show that the multiplicative orders of these matrices do not always coincide with those of permutation matrices of the same size. We pose the problem of identifying finite subgroups of general linear groups that are generated by alternating sign matrices.

On Finite Subgroups in the General Linear Groups over an Algebraic Number Field

As is well-known, there are only finitely many isomorphic classes of finite subgroups in a given general linear group over the field of rational numbers. This result can be generalized to any algebraic number field. While the case of field of rational numbers is relatively well-studied, we still do not know much for general algebraic number field cases. In this article, we discuss the finiteness of isomorphic classes of finite subgroups of general linear groups over an algebraic number field. We give a method to calculate a multiplicative bound for the orders of finite subgroups and to classify finite cyclic subgroups.

C1 actions on the circle of finite index subgroups of Mod(Σg), Aut(Fn), and Out(Fn)

Let Σg be a closed, connected, and oriented surface of genus g≥24, and let Γ be a finite index subgroup of the mapping class group Mod(Σg) that contains the Torelli group ℐ(Σg). Then any orientation-preserving C1 action of Γ on the circle cannot be faithful. We also show that if Γ is a finite index subgroup of Aut(Fn), when n≥8, that contains the subgroup of IA-automorphisms, then any orientation-preserving C1 action of Γ on the circle cannot be faithful. Similarly, if Γ is a finite index subgroup of Out(Fn), when n≥8, that contains the Torelli group 𝒯n, then any orientation preserving C1 action of Γ on the circle cannot be faithful. In fact, when n≥10, any orientation-preserving C1 action of a finite index subgroup of Aut(Fn) or Out(Fn) on the circle cannot be faithful.

Jordan property for automorphism groups of compact spaces in Fujiki's class C$\mathcal {C}$

Let $X$ be a compact complex space in Fujiki's Class $\mathcal{C}$. We show that the group $Aut(X)$ of all biholomorphic automorphisms of $X$ has the Jordan property: there is a (Jordan) constant $J = J(X)$ such that any finite subgroup $G\le Aut(X)$ has an abelian subgroup $H\le G$ with the index $[G:H]\le J$. This extends, with a quite different method, the result of Prokhorov and Shramov for Moishezon threefolds.

Commutativity degree of chains of finite groups

The concepts of commutativity of two chains, and the commutativity degree of the chains of a finite group such as G which ends in G are introduced. Then, the relation between the commutativity degree of the chains of a finite group is examined, and eventually this measure is calculated for some finite groups and fuzzy subgroups.

Entire Vortex Solutions of Negative Degree for the Anisotropic Ginzburg–Landau System

The anisotropic Ginzburg-Landau system \[ \Delta u+\delta\, \nabla (\mathrm{div}\: u) +\delta\, \mathrm{curl}^*(\mathrm{curl}\: u)=(|u|^2-1) u, \] for $u\colon\mathbb R^2\to\mathbb R^2$ and $\delta\in (-1,1)$, models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that $|u(x)|\to 1$ and $u$ has a prescribed topological degree $d\leq -1$ as $|x|\to\infty$, for small values of the anisotropy parameter $|\delta| < \delta_0(d)$. Unlike the isotropic case $\delta=0$, this cannot be reduced to a one-dimensional radial equation. We obtain these solutions by minimizing the anisotropic Ginzburg-Landau energy in an appropriate class of equivariant maps, with respect to a finite symmetry subgroup.

Level structure, arithmetic representations, and noncommutative Siegel linearization

Abstract Let ℓ {\ell} be a prime, k a finitely generated field of characteristic different from ℓ {\ell} , and X a smooth geometrically connected curve over k. Say a semisimple representation of π 1 ét ⁢ ( X k ¯ ) {\pi_{1}^{{\text{\'{e}t}}}(X_{\bar{k}})} is arithmetic if it extends to a finite index subgroup of π 1 ét ⁢ ( X ) {\pi_{1}^{{\text{\'{e}t}}}(X)} . We show that there exists an effective constant N = N ⁢ ( X , ℓ ) {N=N(X,\ell)} such that any semisimple arithmetic representation of π 1 ét ⁢ ( X k ¯ ) {\pi_{1}^{{\text{\'{e}t}}}(X_{\bar{k}})} into GL n ⁡ ( ℤ ℓ ¯ ) {\operatorname{GL}_{n}(\overline{\mathbb{Z}_{\ell}})} , which is trivial mod ℓ N {\ell^{N}} , is in fact trivial. This extends a previous result of the second author from characteristic zero to all characteristics. The proof relies on a new noncommutative version of Siegel’s linearization theorem and the ℓ {\ell} -adic form of Baker’s theorem on linear forms in logarithms.

Brauer characters, degrees and subgroups

We prove a result on Brauer characters of finite groups, subgroups and degrees of characters, obtaining, as a corollary, a shorter proof of a generalization of a recent result of G. Qian on element orders and character degrees.

5D and 6D SCFTs from $\mathbb{C}^3$ orbifolds

We study the orbifold singularities X=\mathbb{C}^3/\GammaX=ℂ3/Γ where \GammaΓ is a finite subgroup of SU(3)SU(3). M-theory on this orbifold singularity gives rise to a 5d SCFT, which is investigated with two methods. The first approach is via 3d McKay correspondence which relates the group theoretic data of \GammaΓ to the physical properties of the 5d SCFT. In particular, the 1-form symmetry of the 5d SCFT is read off from the McKay quiver of \GammaΓ in an elegant way. The second method is to explicitly resolve the singularity XX and study the Coulomb branch information of the 5d SCFT, which is applied to toric, non-toric hypersurface and complete intersection cases. Many new theories are constructed, either with or without an IR quiver gauge theory description. We find that many resolved Calabi-Yau threefolds, \widetilde{X}X̃, contain compact exceptional divisors that are singular by themselves. Moreover, for certain cases of \GammaΓ, the orbifold singularity \mathbb{C}^3/\Gammaℂ3/Γ can be embedded in an elliptic model and gives rise to a 6d (1,0) SCFT in the F-theory construction. Such 6d theory is naturally related to the 5d SCFT defined on the same singularity. We find examples of rank-1 6d SCFTs without a gauge group, which are potentially different from the rank-1 E-string theory.

EQUIVARIANT JACOBIAN CONJECTURE IN DIMENSION TWO

Let G be a small finite subgroup of GL(2,ℂ) and let $$ \overset{\sim }{\varphi }:{\mathbbm{A}}^2\to {\mathbbm{A}}^2 $$ be a G- equivariant étale endomorphism of the affine plane. We show that $$ \overset{\sim }{\varphi } $$ is an automorphism if the order of G is even. The proof depends on an analysis of a quasi-étale endomorphism φ induced by $$ \overset{\sim }{\varphi } $$ on the singular quotient surface $$ {\mathbbm{A}}^2/G $$ whose smooth part X° has the standard $$ {\mathbbm{A}}_{\ast}^1 $$ -fibration p°: X° → ℙ1. If φ preserves the standard $$ {\mathbbm{A}}_{\ast}^1 $$ -fibbration p° then both φ and $$ \overset{\sim }{\varphi } $$ are automorphisms. We look for the condition with which φ preserves the standard $$ {\mathbbm{A}}_{\ast}^1 $$ -fibration and prove as a consequence that $$ \overset{\sim }{\varphi } $$ is an automorphism if G has even order.

On integral representations of finite groups and rings generated by character values

We study realization fields and integrality of characters of finite subgroups of [Formula: see text] and related lattices with a focus on the integrality of characters of finite groups [Formula: see text]. We are interested in the arithmetic aspects of the integral realizability of representations of finite groups, order generated by the character values, the number of minimal realization splitting fields, and in particular, consider the conditions of realizability in the terms of Hilbert symbols and quaternion algebras and some orders generated by character values over the rings of rational and algebraic integers.

Generating ideals by additive subgroups of rings

We obtain several fundamental results on finite index ideals and additive subgroups of rings as well as on model-theoretic connected components of rings, which concern generating in finitely many steps inside additive groups of rings.Let R be any ring equipped with an arbitrary additional first order structure, and A a set of parameters. We show that whenever H is an A-definable, finite index subgroup of (R,+), then H+R⋅H contains an A-definable, two-sided ideal of finite index. As a corollary, we obtain a positive answer to Question 4.9 of [4]: if R is unital, then (R¯,+)A00+R¯⋅(R¯,+)A00+R¯⋅(R¯,+)A00=R¯A00, which also implies that R¯A00=R¯A000, where R¯≻R is a sufficiently saturated elementary extension of R, (R¯,+)A00 [resp. R¯A00] denotes the smallest A-type-definable, bounded index additive subgroup [resp. ideal] of R¯, and R¯A000 is the smallest invariant over A, bounded index ideal of R¯. If R is of positive characteristic (not necessarily unital), we get a sharper result: (R¯,+)A00+R¯⋅(R¯,+)A00=R¯A00. We obtain a similar result (but with more steps required) for finitely generated (not necessarily unital) rings. We obtain analogous results for topological rings. The above result for unital rings implies that the simplified descriptions of the definable (and so also classical) Bohr compactifications of triangular groups over unital rings obtained in Corollary 4.5 of [4] are valid for all unital rings.We also analyze many concrete examples, where we compute the number of steps needed to generate a group by (R¯∪{1})⋅(R¯,+)A00 and study related aspects, showing “optimality” of some of our main results and yielding answers to some natural questions.

Virtual Critical Regularity of Mapping Class Group Actions on the Circle

We show that if G1 and G2 are non-solvable groups, then no C1,τ action of \((G_{1}\times G_{2})*\mathbb {Z}\) on S1 is faithful for τ > 0. As a corollary, if S is an orientable surface of complexity at least three then the critical regularity of an arbitrary finite index subgroup of the mapping class group Mod(S) with respect to the circle is at most one, thus strengthening a result of the first two authors with Baik.

Relative scale separation in orbifolds of S2 and S5

In orbifold vacua containing an Sq/Γ factor, we compute the relative order of scale separation, r, defined as the ratio of the eigenvalue of the lowest-lying Γ-invariant state of the scalar Laplacian on Sq, to the eigenvalue of the lowest-lying state. For q = 2 and Γ finite subgroup of SO(3), or q = 5 and Γ finite subgroup of SU(3), the maximal relative order of scale separation that can be achieved is r = 21 or r = 12, respectively. For smooth S5 orbifolds, the maximal relative scale separation is r = 4.2. Methods from invariant theory are very efficient in constructing Γ-invariant spherical harmonics, and can be readily generalized to other orbifolds.

Formal solutions and the first‐order theory of acylindrically hyperbolic groups

We generalise Merzlyakov's theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if $G$ is an acylindrically hyperbolic group and $E(G)$ denotes the unique maximal finite normal subgroup of $G$, then $G$ and the HNN extension $G\dot{\ast}_{E(G)}$, which is simply the free product $G\ast\mathbb{Z}$ when $E(G)$ is trivial, have the same $\forall\exists$-theory. As a consequence, we prove the following conjecture, formulated by Casals-Ruiz, Garreta and de la Nuez Gonz\'alez: acylindrically hyperbolic groups have trivial positive theory. In particular, one recovers a result proved by Bestvina, Bromberg and Fujiwara, stating that, with only the obvious exceptions, verbal subgroups of acylindrically hyperbolic groups have infinite width.

Automorphic Lie Algebras and Modular Forms

AbstractWe introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma $ be a finite index subgroup of $\textrm {SL}(2,\mathbb Z)$ with an action on a complex simple Lie algebra $\mathfrak g$, which can be extended to $\textrm {SL}(2,{\mathbb {C}})$. We show that the Lie algebra of the corresponding $\mathfrak {g}$-valued modular forms is isomorphic to the extension of $\mathfrak {g}$ over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups $\Gamma (N), \, N\leq 6$, is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras.