Finite Cyclic Group Research Articles

Symmetric Complete Sum-free Sets in Cyclic Groups

We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in [0,13], answering a question of Cameron, and that the number of those contained in the cyclic group of order n is exponential in n. For primes p, we provide a full characterization of the symmetric complete sum-free subsets of Zp of size at least (13−c)⋅p, where c>0 is a universal constant.

The Main Eigenvalues of the Undirected Power Graph of a Group

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one vector. In this paper we want to focus on the power graph of the finite cyclic group $mathbb{Z}_{n}$ and find a condition on n where $P(mathbb{Z}_{n})$ has exactly one main eigenvalue. Then we calculate the number of main eigenvalues of $P(mathbb{Z}_{n})$ where $n$ has a unique prime decomposition $n = p^{r} p_2$. We also formulate a conjecture on the number of the main eigenvalues of $P(mathbb{Z}_{n})$ for an arbitrary positive integer $n$.

Baer Group Rings with Involution

We prove that if a group ring $RG$ is a (quasi) Baer $*$-ring, then so is $R$, whereas converse is not true. Sufficient conditions are given so that for some finite cyclic groups $G$, if $R$ is (quasi-) Baer $*$-ring, then so is the group ring $RG$. We prove that if the group ring $RG$ is a Baer $*$-ring, then so is $RH$ for every subgroup $H$ of $G$. Also, we generalize results of Zhong Yi, Yiqiang Zhou (for (quasi-) Baer rings) and L. Zan, J. Chen (for principally quasi-Baer and principally projective rings).

Efficient dominating sets in circulant graphs

Let S be a subset of finite cyclic group Zn not containing the identity element 0 with S=−S. Cayley graphs on Zn with respect to S are called circulant graphs and denoted by Cay(Zn,S). In this paper, for connected non-complete circulant graphs Cay(Zn,S) of degree |S|=p−1 with p prime, we give a necessary and sufficient condition for the existence of efficient dominating sets, and characterize all efficient dominating sets if exist. We also obtain similar results for Cay(Zn,S) of degree |S|=pq−1 and pm−1, where p,q are primes, m is a positive integer, and |S|+1 is relatively prime to n|S|+1. Moreover, we give a necessary and sufficient condition for the existence of efficient dominating sets in Cay(Zn,S) of order n=pkq,p2q2,pqr,p2qr,pqrs and degree |S|, where p,q,r,s are distinct primes, k is a positive integer, and |S|+1 is relatively prime to n|S|+1.

Formally dual subsets of cyclic groups of prime power order

We study the notion of formal-duality over finite cyclic groups of prime power order as introduced by Cohn, Kumar, Reiher and Sch\"urmann. We will prove that for any cyclic group of odd prime power order, as well as for any cyclic group of order $2^{2l+1}$, there is no primitive pair of formally-dual subsets. This partially proves a conjecture, made by the priorly mentioned authors, that the only cyclic groups with a pair of primitive formally-dual subsets are $\{0\}$ and $\mathbb{Z}/4\mathbb{Z}$.

Free actions on C*-algebra suspensions and joins by finite cyclic groups

We present a proof for certain cases of the noncommutative Borsuk-Ulam conjectures proposed by Baum, D\k{a}browski, and Hajac. When a unital $C^*$-algebra $A$ admits a free action of $\mathbb{Z}/k\mathbb{Z}$, $k \geq 2$, there is no equivariant map from $A$ to the $C^*$-algebraic join of $A$ and the compact "quantum" group $C(\mathbb{Z}/k\mathbb{Z})$. This also resolves D\k{a}browski's conjecture on unreduced suspensions of $C^*$-algebras. Finally, we formulate a different type of noncommutative join than the previous authors, which leads to additional open problems for finite cyclic group actions.

An inverse problem about minimal zero-sum sequences over finite cyclic groups

The article characterizes the minimal zero-sum sequences over the cyclic group Cn with lengths between ⌊n/3⌋+3 and ⌊n/2⌋+1, for n≥10. This is a step beyond established results about minimal zero-sum sequences over Cn of lengths at least ⌊n/2⌋+2. The range of the obtained characterization is optimal.Among the possible approaches we choose one with a strong emphasis on unsplittable sequences—intriguing objects generalizing the longest minimal zero-sum sequences over an abelian group. The unsplittable sequences over Cn with lengths in [⌊n/3⌋+3,⌊n/2⌋+1] prove capable of capturing the essence of our setting and deserve an explicit description.

ON TOPOLOGICAL CLASSIFICATION OF FINITE CYCLIC ACTIONS ON BORDERED SURFACES

In Hirose (Tohoku Math. J. 62 (2010), 45–53), Susumu Hirose showed that, except for a few cases, the order $N$ of a cyclic group of self-homeomorphisms of a closed orientable topological surface $S_{g}$ of genus $g\geqslant 2$ determines the group up to a topological conjugation, provided that $N\geqslant 3g$. Gromadzki et al. undertook in Bagiński et al. (Collect. Math. 67 (2016), 415–429) a more general problem of topological classification of such group actions for $N>2(g-1)$. In Gromadzki and Szepietowski (Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 110 (2016), 303–320), we considered the analogous problem for closed nonorientable surfaces, and in Gromadzki et al. (Pure Appl. Algebra 220 (2016), 465–481) – the problem of classification of cyclic actions generated by an orientation-reversing self-homeomorphism. The present paper, in which we deal with topological classification of actions on bordered surfaces of finite cyclic groups of order $N>p-1$, where $p$ is the algebraic genus of the surface, completes our project of topological classification of ‘‘large” cyclic actions on compact surfaces. We apply obtained results to solve the problem of uniqueness of the actions realizing the solutions of the so-called minimum genus and maximum order problems for bordered surfaces found in Bujalance et al. (Automorphisms Groups of Compact Bordered Klein Surfaces: A Combinatorial Approach, Lecture Notes in Mathematics 1439, Springer, 1990).

Inclusion of configuration spaces in Cartesian products, and the virtual cohomological dimension of the braid groups of 𝕊2 and ℝP2

Let M be a surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface. We consider the inclusion i: F\_n(M) --\textgreater{} M^n of the nth configuration space F\_n(M) of M into the n-fold Cartesian product of M, as well as the induced homomorphism i\_\#: P\_n(M) --\textgreater{} (\pi\_1(M))^n, where P\_n(M) is the n-string pure braid group of M. Both i and i\_\# were studied initially by J.Birman who conjectured that Ker(i\_\#) is equal to the normal closure of the Artin pure braid group P\_n in P\_n(M). The conjecture was later proved by C.Goldberg for compact surfaces without boundary different from the 2-sphere S^2 and the projective plane RP^2. In this paper, we prove the conjecture for S^2 and RP^2. In the case of RP^2, we prove that Ker(i\_\#) is equal to the commutator subgroup of P\_n(RP^2), we show that it may be decomposed in a manner similar to that of P\_n(S^2) as a direct sum of a torsion-free subgroup L\_n and the finite cyclic group generated by the full twist braid, and we prove that L\_n may be written as an iterated semi-direct product of free groups. Finally, we show that the groups B\_n(S^2) and P\_n(S^2) (resp. B\_n(RP^2) and P\_n(RP^2)) have finite virtual cohomological dimension equal to n-3 (resp. n-2), where B\_n(M) denotes the full n-string braid group of M. This allows us to determine the virtual cohomological dimension of the mapping class groups of the mapping class groups of S^2 and RP^2 with marked points, which in the case of S^2, reproves a result due to J.Harer.

THE BOGOMOLOV-PROKHOROV INVARIANT OF SURFACES AS EQUIVARIANT COHOMOLOGY

For a complex smooth projective surface $M$ with an action of a finite cyclic group $G$ we give a uniform proof of the isomorphism between the invariant $H^1(G, H^2(M, {\mathbb Z}))$ and the first cohomology of the divisors fixed by the action, using $G$-equivariant cohomology. This generalizes the main result of Bogomolov and Prokhorov On stable conjugacy of finite subgroups of the plane Cremona group, I.

The actions on the generating functions for the family of the generalized Bernoulli polynomials

In this paper, we study the generalization Bernoulli numbers and polynomials attached to a periodic group homomorphism from a finite cyclic group to the set of complex numbers and derive new periodic group homomorphism by using a fixed periodic group homomorphism. Hence, we obtain not only multiplication formulas, but also some new identities for the generalized Bernoulli polynomials attached to a periodic group homomorphism.

Extremal Bases for Finite Cyclic Groups

For any positive integer $d$, a subset $A$ of ${\mathbb Z}_{m}$ is called a basis of order $d$ for ${\mathbb Z}_{m}$ if every element of ${\mathbb Z}_{m}$ can be written as a sum of at most $d$ not...

Distributivity and pseudocomplementation of lattices of generalized \(L\)-subgroups

The main goal of this paper is to study the lattice of \((0,\mu)\)-\(L\)-subgroups of a group. We characterize abelian groups by the lattice of \((0,\mu)\)-\(L\)-subgroups. Also, we show that a group $G$ is locally cyclic if and only if the lattice of \((0,\mu)\)-\(L\)-subgroups is distributive. As consequence, we obtain that the lattices of all \((\in,\in\vee q)\)-fuzzy subgroups and all fuzzy subgroups of a finite cyclic group are distributive. Finally, we study groups which of the lattice of \((\lambda,\mu)\)-\(L\)-subgroups is pseudocomplemented lattice.

On the dynamics of endomorphisms of finite groups

Aiming at a better understanding of finite groups as finite dynamical systems, we show that by a version of Fitting’s Lemma for groups, each state space of an endomorphism of a finite group is a graph tensor product of a finite directed 1-tree whose cycle is a loop with a disjoint union of cycles, generalizing results of Hernández-Toledo on linear finite dynamical systems, and we fully characterize the possible forms of state spaces of nilpotent endomorphisms via their “ramification behavior”. Finally, as an application, we will count the isomorphism types of state spaces of endomorphisms of finite cyclic groups in general, extending results of Hernández-Toledo on primary cyclic groups of odd order.

A group-invariant version of Lehmer's conjecture on heights

We state and prove a group-invariant version of Lehmer's conjecture on heights, generalizing papers by Zagier (1993) [5] and Dresden (1998) [1] which are special cases of this theorem. We also extend their three cases to a full classification of all finite cyclic groups satisfying the condition that the set of all orbits for which every non-zero element lies on the unit circle is finite and non-empty.

Note on the index conjecture in zero-sum theory and its connection to a Dedekind-type sum

Let S=(a1)⋯(ak) be a minimal zero-sum sequence over a finite cyclic group G. The index conjecture states that if k=4 and gcd⁡(|G|,6)=1, then S has index 1. In this note we study the index conjecture and connect it to a Dedekind-type sum. In particular we reprove a special case of the conjecture when |G| is prime.

Fractional integration operator on some radial rays and intertwining for the Dunkl operator

Abstract In this paper we consider the differential-difference reflection operator associated with a finite cyclic group, $$Y_\nu f(x)=\frac{df(x)}{dx}+\sum_{i=1}^{m-1}\frac{m\nu_i+m-i}{x}\sum_{j=0}^{m-1}\varepsilon^{-ij}f(\varepsilon^jx).$$ Y ν f ( x ) = d f ( x ) d x + ∑ i = 1 m − 1 m ν i + m − i x ∑ j = 0 m − 1 ε − i j f ( ε j x ) . First we show that the Dimovski ([5], [6]) hyper–Bessel differential operator of arbitrary integer order m is close in frame of the algebra similar to U(sl(2;C)). Secondly, we introduce a difference-differential operator associated to finite cyclic group in the rank one case, and then by using a Poisson-type integral transform proposed by Dimovski and Kiryakova ([7], [11]), we construct a new explicit intertwining (transmutation) operator between the operator Yν and the derivative operator d/dx. It is to emphasize that both hyper–Bessel operators and the so-called Poisson–Dimovski transformation (transmutation) are typical examples of the operators of generalized fractional calculus [11, 12].

Long unsplittable zero-sum sequences over a finite cyclic group

Let [Formula: see text] be an additively written finite cyclic group of order [Formula: see text] and let [Formula: see text] be a minimal zero-sum sequence with elements of [Formula: see text], i.e. the sum of elements of [Formula: see text] is zero, but no proper nontrivial subsequence of [Formula: see text] has sum zero. [Formula: see text] is called unsplittable if there do not exist an element [Formula: see text] in [Formula: see text] and two elements [Formula: see text] in [Formula: see text] such that [Formula: see text] and the new sequence [Formula: see text] is still a minimal zero-sum sequence. In this paper, we investigate long unsplittable minimal zero-sum sequences over [Formula: see text]. Our main result characterizes the structures of all such sequences [Formula: see text] and shows that the index of [Formula: see text] is at most 2, provided that the length of [Formula: see text] is greater than or equal to [Formula: see text] where [Formula: see text] is a positive integer with least prime divisor greater than [Formula: see text].

Hyperbolic rotations about links in 3-manifolds

In this paper, we will show that for any link L in a closed orientable 3-manifold M, infinitely many hyperbolic 3-manifolds are obtained from M by Dehn surgeries so that each of them admits an orientation-preserving smooth finite cyclic group action with fixed point set L.

On Finite Nonabelian Loop Capable Groups

We say that groups, which are isomorphic to inner mapping groups of finite loops, are loop capable. Let p and q be distinct prime numbers, S a nonabelian group of order pq, and C a finite nontrivial cyclic group such that gcd (|S|, |C|) = 1. We show that the group S × C is not loop capable.