Finite Abelian Group Research Articles

Categorification of integral group rings extended by one dimension

AbstractThe integral group rings for finite groups are precisely those fusion rings whose basis elements have Frobenius–Perron dimension 1, and each is categorifiable in the sense that it arises as the Grothendieck ring of a fusion category. Here, we analyze the structure and representation theory of fusion rings with a basis of elements whose Frobenius–Perron dimensions take exactly one value distinct from 1. Our goal is a set of results to assist in characterizing when such fusion rings are categorifiable. As proof of concept, we complete the classification of categorifiable near‐group fusion rings for an infinite collection of finite abelian groups, a task that to‐date has only been completed for three such groups.

Effective Projections on Group Shifts to Decide Properties of Group Cellular Automata

Many decision problems concerning cellular automata are known to be decidable in the case of algebraic cellular automata, that is, when the state set has an algebraic structure and the automaton acts as a morphism. The most studied cases include finite fields, finite commutative rings and finite commutative groups. In this paper, we provide methods to generalize these results to the broader case of group cellular automata, that is, the case where the state set is a finite (possibly non-commutative) finite group. The configuration space is not even necessarily the full shift but a subshift — called a group shift — that is a subgroup of the full shift on [Formula: see text], for any number [Formula: see text] of dimensions. We show, in particular, that injectivity, surjectivity, equicontinuity, sensitivity and nilpotency are decidable for group cellular automata, and non-transitivity is semi-decidable. Injectivity always implies surjectivity, and jointly periodic points are dense in the limit set. The Moore direction of the Garden-of-Eden theorem holds for all group cellular automata, while the Myhill direction fails in some cases. The proofs are based on effective projection operations on group shifts that are, in particular, applied on the set of valid space-time diagrams of group cellular automata. This allows one to effectively construct the traces and the limit sets of group cellular automata. A preliminary version of this work was presented at the conference Mathematical Foundations of Computer Science 2020.

Coarse Sheaf Cohomology

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0, they see the number of ends of the space. In this paper, a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition, coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus, we can use topological tools on compact Hausdorff spaces in our computations. In particular, if the asymptotic dimension of a proper metric space is finite, then higher cohomology groups vanish. We compute a few examples. As it turns out, finite abelian groups are best suited as coefficients on finitely generated groups.

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.

On isomorphisms between ideals of Fourier algebras of finite abelian groups

On isomorphisms between ideals of Fourier algebras of finite abelian groups

Cohomology of Graded Twisting of Hopf Algebras

Let A be a Hopf algebra and B a graded twisting of A by a finite abelian group Γ. Then, categories of comodules over A and B are equivalent (but they are not necessarily monoidally equivalent). We show the relation between the Hochschild cohomology of A and B explicitly. This partially answer a question raised by Bichon. As an application, we prove that A is a twisted Calabi–Yau Hopf algebra if and only if B is a twisted Calabi–Yau algebra, and give the relation between their Nakayama automorphisms.

On the Structure of the Mislin Genus of a Pullback

The notion of genus for finitely generated nilpotent groups was introduced by Mislin. Two finitely generated nilpotent groups Q and R belong to the same genus set G(Q) if and only if the two groups are nonisomorphic, but for each prime p, their p-localizations Qp and Rp are isomorphic. Mislin and Hilton introduced the structure of a finite abelian group on the genus if the group Q has a finite commutator subgroup. In this study, we consider the class of finitely generated infinite nilpotent groups with a finite commutator subgroup. We construct a pullback Ht from the l-equivalences Hi→H and Hj→H, t≡(i+j)mods, where s=|G(H)|, and compare its genus to that of H. Furthermore, we consider a pullback L of a direct product G×K of groups in this class. Here, we prove results on the group L and prove that its genus is nontrivial.

An inequality for non-microstates free entropy dimension for crossed products by finite abelian groups

For certain generating sets of the subfactor pair M\subset M\rtimes G where G is a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Schreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of the form M\rtimes(\mathbb{Z}/2\mathbb{Z})^{\oplus\infty} for a large class of algebras M .

Characterizations of B-valued concentration inequalities via the Rademacher type

In this paper, we show that a sufficient and necessary condition for that the Hoeffding concentration inequality of Banach space-valued ([Formula: see text]-valued, for simplicity) random variables holds is that the Banach space in question admits the Rademacher type [Formula: see text] for some [Formula: see text]; which is equivalent to that the Bernstein concentration inequality holds for such [Formula: see text]-valued random variables. This is done by applying a refined McDiarmid inequality (stated and proved in this paper) via the entropy method. This result is also applied to the density property of induced subgraphs of random Cayley graphs generated by a finite abelian group.

Perfect state transfer on weighted bi-Cayley graphs over abelian groups

The study of perfect state transfer on graphs has attracted extensive attention because of its application in quantum information and computation. This paper establishes necessary and sufficient conditions for a weighted bi-Cayley graph having perfect state transfer over any given finite abelian group. As special cases of weighted bi-Cayley graphs, the results in this paper can apply to examine weighted Cayley graphs having perfect state transfer over dihedral groups, generalized dihedral groups, semi-dihedral groups and dicyclic groups.

Complete factorizations of finite abelian groups

Let G be an abelian group and A1,A2,…,Ak(k≥2) be nonempty subsets of G. The sets A1,A2,…,Ak are said to form a complete factorization of order k of G if A1,A2,…,Ak are mutually disjoint and each g∈G can be uniquely expressed as the sum of a1,a2,…,ak with ai∈Ai for each i=1,2,…,k. If G is finite, then the size of a complete factorization A1,A2,…,Ak of G is defined to be ∑i=1k|Ai|. In this paper, we characterize finite abelian groups that have a complete factorization. We also determine the minimum size of a complete factorization of finite abelian groups.

A recursive formula for the product of element orders of finite abelian groups

Let G be a finite group and let ψ(G) denote the sum of element orders of G; later this concept has been used to define R(G) which is the product of the element orders of G. Motivated by the recursive formula for ψ(G), we consider a finite abelian group G and obtain a similar formula for R(G).

Crossed Product Equivalence of Quantum Automorphism Groups of Finite Dimensional C*-Algebras

Abstract We compare the algebras of the quantum automorphism group of finite-dimensional C$^\ast $-algebra $B$, which includes the quantum permutation group $S_N^+$, where $N = \dim B$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $\Gamma $ lead to isomorphic $\ast $-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $1$-boundedness between the various algebras associated with these quantum groups.

Elliptic Curves of Type y2=x3−3pqx Having Ranks Zero and One

The group of rational points on an elliptic curve over Q is always a finitely generated Abelian group, hence isomorphic to Zr×G with G a finite Abelian group. Here, r is the rank of the elliptic curve. In this paper, we determine sufficient conditions that need to be set on the prime numbers p and q so that the elliptic curve E:y2=x3−3pqx over Q would possess a rank zero or one. Specifically, we verify that if distinct primes p and q satisfy the congruence p≡q≡5(mod24), then E has rank zero. Furthermore, if p≡5(mod12) is considered instead of a modulus of 24, then E has rank zero or one. Lastly, for primes of the form p=24k+17 and q=24ℓ+5, where 9k+3ℓ+7 is a perfect square, we show that E has rank one.

Inverse problems associated with subsequence sums in Cp ⊕ Cp II

Let G be a finite abelian group and S a sequence with elements of G. Let Σk(S)⊂G denote the set of group elements which can be expressed as a sum of a nonempty subsequence of S with length k. In this paper, we study the set Σp2(S) of the sequence S over Cp⊕Cp, where p is a prime and |S|=p2+r with p−1≤r≤2p−3. We prove that either 0∈Σp2(S) or |Σp2(S)|≥p(r−p+3)−1; furthermore, we determine the structure of S if 0∉Σp2(S) and |Σp2(S)|=p(r−p+3)−1. These results confirm some cases of two conjectures on p2-sums of a sequence over Cp⊕Cp.

Monomial projections of Veronese varieties: New results and conjectures

In this paper, we consider the homogeneous coordinate rings A(Yn,d)≅K[Ωn,d] of monomial projections Yn,d of Veronese varieties parameterized by subsets Ωn,d of monomials of degree d in n+1 variables where: (1) Ωn,d contains all monomials supported in at most s variables and, (2) Ωn,d is a set of monomial invariants of a finite diagonal abelian group G⊂GL(n+1,K) of order d. Our goal is to study when K[Ωn,d] is a quadratic algebra and, if so, when K[Ωn,d] is Koszul or G-quadratic. For the family (1), we prove that K[Ωn,d] is quadratic when s≥⌈n+22⌉. For the family (2), we completely characterize when K[Ω2,d] is quadratic in terms of the group G⊂GL(3,K), and we prove that K[Ω2,d] is quadratic if and only if it is Koszul. We also provide large families of examples where K[Ωn,d] is G-quadratic.

On Davenport Constant of the Group $C_2^{r-1} \oplus C_{2k}$

Let $G$ be a finite abelian group. The Davenport constant $\mathsf{D}(G)$ is the maximal length of minimal zero-sum sequences over $G$. For groups of the form $C_2^{r-1} \oplus C_{2k}$ the Davenport constant is known for $r\leq 5$. In this paper, we get the precise value of $\mathsf{D}(C_2^{5} \oplus C_{2k})$ for $k\geq 149$. It is also worth pointing out that our result can imply the precise value of $\mathsf{D}(C_2^{4} \oplus C_{2k})$.

Finite Abelian Groups via Congruences

For every finite abelian group G, there are positive integers n and d such that G is isomorphic to the multiplicative group of d-th powers of reduced residues modulo n.

Perfect state transfer on bi-Cayley graphs over abelian groups

The study of perfect state transfer on graphs has attracted a great deal of attention during the past ten years because of its applications to quantum information processing and quantum computation. Perfect state transfer is understood to be a rare phenomenon. This paper establishes necessary and sufficient conditions for a bi-Cayley graph having a perfect state transfer over any given finite abelian group. As corollaries, many known and new results are obtained on Cayley graphs having perfect state transfer over abelian groups, (generalized) dihedral groups, semi-dihedral groups and generalized quaternion groups. Especially, we give an example of a connected non-normal Cayley graph over a dihedral group having perfect state transfer between two distinct vertices, which was thought impossible.

Level structures on p-divisible groups from the Morava E-theory of abelian groups

The close relationship between the scheme of level structures on the universal deformation of a formal group and the Morava E-cohomology of finite abelian groups has played an important role in the study of power operations for Morava E-theory. The goal of this paper is to explore the relationship between level structures on the p-divisible group given by the trivial extension of the universal deformation by a constant p-divisible group and the Morava E-cohomology of the iterated free loop space of the classifying space of a finite abelian group.