Cyclic Group Of Prime Order Research Articles

The structure of arbitrary Conze–Lesigne systems

Let Γ \Gamma be a countable abelian group. An (abstract) Γ \Gamma -system X \mathrm {X} - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of Γ \Gamma - is said to be a Conze–Lesigne system if it is equal to its second Host–Kra–Ziegler factor Z 2 ( X ) \mathrm {Z}^2(\mathrm {X}) . The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian Γ \Gamma , namely that they are the inverse limit of translational systems G n / Λ n G_n/\Lambda _n arising from locally compact nilpotent groups G n G_n of nilpotency class 2 2 , quotiented by a lattice Λ n \Lambda _n . Results of this type were previously known when Γ \Gamma was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers U 3 ( G ) U^3(G) norm for arbitrary finite abelian groups G G .

Fixed point sets and orbit spaces of wedge of three spheres

Let X be a finite CW-complex having mod p cohomology isomorphic to a wedge of three spheres Sn∨Sm∨Sl,1≤n≤m≤l. The aim of this paper is to determine the fixed point sets of actions of the cyclic group of prime order on X. We also classify the orbit spaces of free actions of G=Zp,p a prime or G=Sd,d=1 or 3, on X and derive the Borsuk-Ulam type results.

A note on the degree bounds of the invariant ring

<abstract><p>Let $ G = C_p\times H $ be a finite group, where $ C_p $ is a cyclic group of prime order $ p $ and $ H $ is a $ p^{\prime} $-group. Let $ \mathbb{F} $ be an algebraically closed field in characteristic $ p $. Let $ V $ be a direct sum of $ m $ non-trivial indecomposable $ G $-modules such that the norm polynomials of the simple $ H $-modules are the power of the product of the basis elements of the dual. In previous work, we proved the periodicity property of the polynomial ring $ \mathbb{F}[V] $ with actions of $ G $. In this paper, by the periodicity property, we showed that $ \mathbb{F}[V]^G $ is generated by $ m $ norm polynomials together with homogeneous invariants of degree at most $ m|G|-{\rm dim}(V) $ and transfer invariants, which yields the well-known degree bound $ {\rm dim} $$ (V)\cdot(|G|-1) $. More precisely, we found that this bound gets less sharp as the dimensions of simple $ H $-modules increase.</p></abstract>

𝑝-group Galois covers of curves in characteristic 𝑝

We study cohomologies of a curve with an action of a finite p p -group over a field of characteristic p p . Assuming the existence of a certain “magical element” in the function field of the curve, we compute the equivariant structure of the module of holomorphic differentials and the de Rham cohomology, up to certain local terms. We show that a generic p p -group cover has a “magical element”. As an application we compute the de Rham cohomology of a curve with an action of a finite cyclic group of prime order.

Class numbers, cyclic simple groups, and arithmetic

AbstractHere, we initiate a program to study relationships between finite groups and arithmetic–geometric invariants in a systematic way. To do this, we first introduce a notion of optimal module for a finite group in the setting of holomorphic mock Jacobi forms. Then, we classify optimal modules for the cyclic groups of prime order, in the special case of weight 2 and index 1, where class numbers of imaginary quadratic fields play an important role. Finally, we exhibit a connection between the classification we establish and the arithmetic geometry of imaginary quadratic twists of modular curves of prime level.

CI-property of [formula omitted] and [formula omitted] for digraphs

We prove that the direct product of two coprime order elementary abelian groups of rank two, as well as the direct product of a cyclic group of prime order and a cyclic group of square-free order are DCI-groups. The latter is a generalization of Muzychuk's result on cyclic groups (J. Combin. Theory Ser. A, 1995).

Constructions of exotic actions on product manifolds with an asymmetric factor

We explore transformation groups of manifolds of the form M×Sn, where M is an asymmetric manifold, that is, a manifold which does not admit any nontrivial action of a finite group. In particular, we prove that for n=2, there exists an infinite family of distinct nondiagonal effective circle actions on such products. A similar result holds for actions of cyclic groups of prime order. We also discuss free circle actions on M×S1, where M belongs to the class of “almost asymmetric” manifolds considered previously by Puppe and Kreck.

Determination of compact Lie groups with the Borsuk-Ulam property

A compact Lie group G is said to have the Borsuk-Ulam property if the Borsuk-Ulam theorem holds for G-maps between representation spheres. It is well-known that an elementary abelian p-group Cpn (p any prime) and an n-torus Tn have the Borsuk-Ulam property. In this paper, we shall discuss the classical question of which compact Lie groups have the Borsuk-Ulam property and in particular we shall show that every extension group of an n-torus by a cyclic group of prime order does not have the Borsuk-Ulam property. This leads us that the only compact Lie groups with the Borsuk-Ulam property are Cpn and Tn, which is a final answer to the question.

Split logarithm problem and a candidate for a post-quantum signature scheme

A new form of the hidden discrete logarithm problem, called split logarithm problem, is introduced as primitive of practical post-quantum digital signature schemes, which is characterized in using two non-permutable elements $A$ and $B$ of a finite non-commutative associative algebra, which are used to compute generators $Q=AB$ and $G=BQ$ of two finite cyclic groups of prime order $q$. The public key is calculated as a triple of vectors $(Y,Z,T)$: $Y=Q^x$, $Z=G^w$, and $T=Q^aB^{-1}G^b$, where $x$, $w$, $a$, and $b$ are random integers. Security of the signature scheme is defined by the computational difficulty of finding the pair of integers $(x,w)$, although, using a quantum computer, one can easily find the ratio $x/w\bmod q$.

Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds

Abstract We prove a general form of the regularity theorem for uniformity norms, and deduce an inverse theorem for these norms which holds for a class of compact nilspaces including all compact abelian groups, and also nilmanifolds; in particular we thus obtain the first non-abelian versions of such theorems. We derive these results from a general structure theorem for cubic couplings, thereby unifying these results with the Host–Kra Ergodic Structure Theorem. A unification of this kind had been propounded as a conceptual prospect by Host and Kra. Our work also provides new results on nilspaces. In particular, we obtain a new stability result for nilspace morphisms. We also strengthen a result of Gutman, Manners and Varjú, by proving that a k-step compact nilspace of finite rank is a toral nilspace (in particular, a connected nilmanifold) if and only if its k-dimensional cube set is connected. We also prove that if a morphism from a cyclic group of prime order into a compact finite-rank nilspace is sufficiently balanced (i.e. equidistributed in a certain quantitative and multidimensional sense), then the nilspace is toral. As an application of this, we obtain a new proof of a refinement of the Green–Tao–Ziegler inverse theorem.

Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group II

Let G be an additive finite abelian group. Let F(G) denote the monoid of all sequences over G. For Ω⊂F(G), let dΩ(G) be the smallest integer t such that every sequence S over G of length |S|≥t has a subsequence in Ω. A sequence S over G is a weak-regular sequence if vg(S)≤ord(g) for every g∈G. Let N(G) be the smallest integer t such that every weak-regular sequence S over G of length |S|≥t has a nontrivial zero-sum subsequence T with vg(T)=vg(S)>0 for some g|S. The invariant N(G) was formulated in a recent paper [4] to study dΩ(G) and has its own interest, and N(G) has been determined for the cyclic groups of prime order there. In this paper, among other results, we shall determine the precise value of N(G) for cyclic groups of prime power order and for elementary abelian groups of rank two.

CGP17Pat: Automated Schizophrenia Detection Based on a Cyclic Group of Prime Order Patterns Using EEG Signals.

Background and Purpose: Machine learning models have been used to diagnose schizophrenia. The main purpose of this research is to introduce an effective schizophrenia hand-modeled classification method. Method: A public electroencephalogram (EEG) signal data set was used in this work, and an automated schizophrenia detection model is presented using a cyclic group of prime order with a modulo 17 operator. Therefore, the presented feature extractor was named as the cyclic group of prime order pattern, CGP17Pat. Using the proposed CGP17Pat, a new multilevel feature extraction model is presented. To choose a highly distinctive feature, iterative neighborhood component analysis (INCA) was used, and these features were classified using k-nearest neighbors (kNN) with the 10-fold cross-validation and leave-one-subject-out (LOSO) validation techniques. Finally, iterative hard majority voting was employed in the last phase to obtain channel-wise results, and the general results were calculated. Results: The presented CGP17Pat-based EEG classification model attained 99.91% accuracy employing 10-fold cross-validation and 84.33% accuracy using the LOSO strategy. Conclusions: The findings and results depicted the high classification ability of the presented cryptologic pattern for the data set used.

Integral cohomology of quotients via toric geometry

We describe the integral cohomology of $X/G$ where $X$ is a compact complex manifold and $G$ a cyclic group of prime order with only isolated fixed points. As a preliminary step, we investigate the integral cohomology of toric blow-ups of quotients of $\mathbb{C}^n$. We also provide necessary and sufficient conditions for the spectral sequence of equivariant cohomology of $(X,G)$ to degenerate at the second page. As an application, we compute the Beauville--Bogomolov form of $X/G$ when $X$ is a Hilbert scheme of points on a K3 surface and $G$ a symplectic automorphism group of orders 5 or 7.

The spectrum of equivariant Kasparov theory for cyclic groups of prime order

We compute the Balmer spectrum of the equivariant bootstrap category of separable $G$-C*-algebras when $G$ is a group of prime order.

On the classification of group actions on C*-algebras up to equivariant KK-equivalence

We study the classification of group actions on C*-algebras up to equivariant KK-equivalence. We show that any group action is equivariantly KK-equivalent to an action on a simple, purely infinite C*-algebra. We show that a conjecture of Izumi is equivalent to an equivalence between cocycle conjugacy and equivariant KK-equivalence for actions of torsion-free amenable groups on Kirchberg algebras. Let G be a cyclic group of prime order. We describe its actions up to equivariant KK-equivalence, based on previous work by Manuel K\"ohler. In particular, we classify actions of G on stabilised Cuntz algebras in the equivariant bootstrap class up to equivariant KK-equivalence.

Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules

Linear molecules usually represent a special case in rotational-vibrational calculations due to a singularity of the kinetic energy operator that arises from the rotation about the a (the principal axis of least moment of inertia, becoming the molecular axis at the linear equilibrium geometry) being undefined. Assuming the standard ro-vibrational basis functions, in the 3N−6 approach, of the form ∣ν1,ν2,ν3ℓ3;J,k,m⟩, tackling the unique difficulties of linear molecules involves constraining the vibrational and rotational functions with k=ℓ3, which are the projections, in units of ℏ, of the corresponding angular momenta onto the molecular axis. These basis functions are assigned to irreducible representations (irreps) of the C2v(M) molecular symmetry group. This, in turn, necessitates purpose-built codes that specifically deal with linear molecules. In the present work, we describe an alternative scheme and introduce an (artificial) group that ensures that the condition ℓ3=k is automatically applied solely through symmetry group algebra. The advantage of such an approach is that the application of symmetry group algebra in ro-vibrational calculations is ubiquitous, and so this method can be used to enable ro-vibrational calculations of linear molecules in polyatomic codes with fairly minimal modifications. To this end, we construct a—formally infinite—artificial molecular symmetry group D∞h(AEM), which consists of one-dimensional (non-degenerate) irreducible representations and use it to classify vibrational and rotational basis functions according to ℓ and k. This extension to non-rigorous, artificial symmetry groups is based on cyclic groups of prime-order. Opposite to the usual scenario, where the form of symmetry adapted basis sets is dictated by the symmetry group the molecule belongs to, here the symmetry group D∞h(AEM) is built to satisfy properties for the convenience of the basis set construction and matrix elements calculations. We believe that the idea of purpose-built artificial symmetry groups can be useful in other applications.

The Segal conjecture for topological Hochschild homology of Ravenel spectra

In the 1980’s, Ravenel introduced sequences of spectra X(n) and T(n) which played an important role in the proof of the Nilpotence Theorem of Devinatz–Hopkins–Smith. In the present paper, we solve the homotopy limit problem for topological Hochschild homology of X(n), which is a generalized version of the Segal Conjecture for the cyclic groups of prime order. This result is the first step towards computing the algebraic K-theory of X(n) using trace methods, which approximates the algebraic K-theory of the sphere spectrum in a precise sense. We solve the homotopy limit problem for topological Hochschild homology of T(n) under the assumption that the canonical map \(T(n)\rightarrow BP\) of homotopy commutative ring spectra can be rigidified to map of \(E_2\) ring spectra. We show that the obstruction to our assumption holding can be described in terms of an explicit class in an Atiyah-Hirzebruch spectral sequence.

Generalized Hadamard Matrices and 2-Factorization of Complete Graphs

Graph factorization plays a major role in graph theory and it shares common ideas in important problems such as edge coloring and Hamiltonian cycles. A factor of a graph is a spanning subgraph of which is not totally disconnected. An - factor is an - regular spanning subgraph of and is -factorable if there are edge-disjoint -factors such that . We shall refer as an -factorization of a graph . In this research we consider -factorization of complete graph. A graph with vertices is called a complete graph if every pair of distinct vertices is joined by an edge and it is denoted by . We look into the possibility of factorizing with added limitations coming in relation to the rows of generalized Hadamard matrix over a cyclic group. Over a cyclic group of prime order , a square matrix of order all of whose elements are the root of unity is called a generalized Hadamard matrix if , where is the conjugate transpose of matrix and is the identity matrix of order . In the present work, generalized Hadamard matrices over a cyclic group have been considered. We prove that the factorization is possible for in the case of the limitation 1, namely, If an edge belongs to the factor , then the and entries of the corresponding generalized Hadamard matrix should be different in the row. In Particular, number of rows in the generalized Hadamard matrices is used to form -factorization of complete graphs. We discuss some illustrative examples that might be used for studying the factorization of complete graphs.

Some new results about a conjecture by Brian Alspach

In this paper, we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset A of mathbb {Z}_n{setminus } {0} of size k such that sum _{zin A} znot = 0, it is possible to find an ordering (a_1,ldots ,a_k) of the elements of A such that the partial sums s_i=sum _{j=1}^i a_j, i=1,ldots ,k, are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size kle 11 in cyclic groups of prime order. Here, we extend this result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in mathbb {Z}_n. We also consider a related conjecture, originally proposed by Ronald Graham: given a subset A of mathbb {Z}_p{setminus }{0}, where p is a prime, there exists an ordering of the elements of A such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis, and Schmitt, based on Alon’s combinatorial Nullstellensatz, we prove the validity of this conjecture for subsets A of size 12.

The Weighted Davenport Constant of a Group and a Related Extremal Problem

For a finite abelian group $G$ written additively, and a non-empty subset $A\subset [1,\exp(G)-1]$ the weighted Davenport Constant of $G$ with respect to the set $A$, denoted $D_A(G)$, is the least positive integer $k$ for which the following holds: Given an arbitrary sequence $(x_1,\ldots,x_k)$ with $x_i\in G$, there exists a non-empty subsequence $(x_{i_1},\ldots,x_{i_t})$ along with $a_{j}\in A$ such that $\sum_{j=1}^t a_jx_{i_j}=0$. In this paper, we pose and study a natural new extremal problem that arises from the study of $D_A(G)$: For an integer $k\ge 2$, determine $f^{(D)}_G(k):=\min\{|A|: D_A(G)\le k\}$ (if the problem posed makes sense). It turns out that for $k$ 'not-too-small', this is a well-posed problem and one of the most interesting cases occurs for $G=\mathbb{Z}_p$, the cyclic group of prime order, for which we obtain near optimal bounds for all $k$ (for sufficiently large primes $p$), and asymptotically tight (up to constants) bounds for $k=2,4$.