Narrow Results

In this paper, the concept of $\mu F$-subgroup is introduced. It is proved that in finite cyclic groups ${{𝒞}}_n$, the collection $L\mu F({{𝒞}}_n)$ of all $\mu F$-subgroups of ${{𝒞}}_n$ is a sublattice of $L({{𝒞}}_n)$. It is also proved that $L\mu F({{𝒞}}_m\times {{𝒞}}_n)$ is not a sublattice of $L({{𝒞}}_m\times {{𝒞}}_n)$ in general. Further, a characterization is obtained for $L\mu F({{𝒞}}_m\times {{𝒞}}_n)$ to be a sublattice of $L({{𝒞}}_m\times {{𝒞}}_n)$.

The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of $p$-groups are studied. In particular, the number of components of that of abelian $p$-groups are determined.

A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=a-b$ of some elements $a,b\in B$. The smallest cardinality $\left|B\right|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $\mathrm{\Delta }[G]$. We prove that for every $n\in \mathbb{N}$ the cyclic group $C_n$ of order $n$ has difference size $\frac{1+\sqrt{4\left|n\right|-3}}{2}\le \mathrm{\Delta }[C_n]\le \frac{3}{2}\sqrt{n}$. If $n\ge 9$ (and $n\ge 2\cdot 10^{15}$), then $\mathrm{\Delta }[C_n]\le \frac{12}{\sqrt{73}}\sqrt{n}$ (and $\mathrm{\Delta }[C_n]<\frac{2}{\sqrt{3}}\sqrt{n}$). Also, we calculate the difference sizes of all cyclic groups of cardinality $\le 100$.

The power graph ${𝒫}(G)$ of a given finite group $G$ is the simple undirected graph whose vertices are the elements of $G$, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity $\kappa ({𝒫}(G))$ of ${𝒫}(G)$ is the minimum number of vertices which need to be removed from $G$ so that the induced subgraph of ${𝒫}(G)$ on the remaining vertices is disconnected or has only one vertex. For a positive integer $n$, let $C_n$ be the cyclic group of order $n$. Suppose that the prime power decomposition of $n$ is given by $n=p_1^{n_1}{p}_2^{n_2}\cdots p_r^{n_r}$, where $r\ge 1$, $n_1,n_2,\dots ,n_r$ are positive integers and $p_1,p_2,\dots ,p_r$ are prime numbers with $p_1<p_2<\cdots <p_r$. The vertex connectivity $\kappa ({𝒫}(C_n))$ of ${𝒫}(C_n)$ is known for $r\le 3$, see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl.17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for $r\ge 4$, we give a new upper bound for $\kappa ({𝒫}(C_n))$ and determine $\kappa ({𝒫}(C_n))$ when $n_r\ge 2$. We also determine $\kappa ({𝒫}(C_n))$ when $n$ is a product of distinct prime numbers.

We retract [1, Lemma 3.4] as the statement is incorrect. In consequence, we correct the statements of Theorems 1.2 and 4.5 and their proofs.

The power graph ${𝒫}(G)$ of a finite group $G$ is the undirected simple graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer $n\ge 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the number of distinct prime divisors of $n$. The minimum degree $\delta ({𝒫}(C_n))$ of ${𝒫}(C_n)$ is known for $r\in \{1,2\}$, see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For $r\ge 3$, under certain conditions involving the prime divisors of $n$, we identify at most $r-1$ vertices such that $\delta ({𝒫}(C_n))$ is equal to the degree of at least one of these vertices. If $r=3$, or that $n$ is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of $n$.

The power graph ${𝒫}(G)$ of a finite group $G$ is the simple graph with vertex set $G$, in which two distinct vertices are adjacent if one of them is a power of the other. For an integer $n\ge 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the number of distinct prime divisors of $n$. For $r\le 3$, the minimum cut-sets of ${𝒫}(C_n)$ are characterized in [S. Chattopadhyay, K. L. Patra and B. K. Sahoo, Vertex connectivity of the power graph of a finite cyclic group, Discrete Appl. Math.266 (2019) 259–271]. In this paper, for $r\ge 4$, we identify certain cut-sets of ${𝒫}(C_n)$ such that any minimum cut-set of ${𝒫}(C_n)$ must be one of them.

Let G be a finite cyclic group. Every sequence S over G can be written in the form S = (n₁g)⋅…⋅(n_lg) where g ∈ G and n₁,…,n_l ∈ [1, ord(g)], and the index ind(S) of S is defined to be the minimum of (n₁ +⋯+ n_l)/ord(g) over all possible g ∈ G such that 〈g〉 = G. A conjecture says that if G is finite such that gcd(|G|, 6) = 1, then ind(S) = 1 for every minimal zero-sum sequence S. In this paper, we prove that the conjecture holds if S is reduced and at least one n_i coprime to |G|.

Let ℤ_m be the group of residue classes modulo m. Let s(m, n) denote the total number of subgroups of the group ℤ_m × ℤ_n, where m and n are arbitrary positive integers. We derive asymptotic formulas for the sum ∑_m,n≤x s(m, n) and for the corresponding sum restricted to gcd(m, n) > 1 which concerns the groups ℤ_m × ℤ_n having rank two.

Let G be a finite cyclic group. Every sequence S over G can be written in the form S = (n₁g) ⋅ … ⋅ (n_lg) where g ∈ G and n₁, …, n_l ∈ [1, ord(g)], and the index ind S of S is defined to be the minimum of (n₁ + ⋯ + n_l)/ord(g) over all possible g ∈ G such that 〈g〉 = G. A conjecture says that if G is finite such that gcd(|G|,6) = 1, then ind(S) = 1 for every minimal zero-sum sequence S. In this paper, we prove that the conjecture holds if S is reduced and the (A1) condition is satisfied (see [On the index-conjecture of length four minimal zero-sum sequences, Int. J. Number Theory9(6) (2013) 1505–1528]).

Let $G$ be a finite groups. The power graph of $G$ is a graph with vertex set $G$ and two distinct element $x,y$ are adjacent if one of them is power another. We give a short elementary proof for $\mathrm{\Gamma }({\mathbb{Z}}_n)$ has a maximum number edges in all groups of order $n$.

Let ${𝒫}(G)$ denote the power graph of a finite group $G$. Let $\rho (Q({𝒫}({\mathbb{Z}}_n)))$ denote the Signless Laplacian spectral radius of ${𝒫}({\mathbb{Z}}_n)$. In this paper, we give lower and upper bounds on $\rho (Q({𝒫}({\mathbb{Z}}_n)))$ for any $n>2$ and find those graphs for which the extremal values are attained. We give a comparison between the bounds obtained and exact value of $\rho (Q({𝒫}({\mathbb{Z}}_n)))$ for any $n>2$. We then find the eigenvalues of $Q({𝒫}(D_n))$ and give lower and upper bounds on the spectral radius of $Q({𝒫}(D_n))$. When $n=p^m$ and $n=pq,$ where $p,q$ are primes and $m$ is a positive integer, we obtain sharper bounds on $\rho (Q({𝒫}(D_n)))$. Finally, we make a conjecture on $\rho (Q({𝒫}(D_n)))$ for any $n>2$.

In this paper, we consider the prime square order Cayley graph of a cyclic group $G$ of order $p^2{q}^2$, where $p$ and $q$ are two distinct prime numbers. We discuss the Eulerianity and Hamiltonicity of the graph. Moreover, we present the girth, the diameter, the independence number, the chromatic number, the clique number, and the dominating number of the graph.

In this paper, we study the cyclic inverse monoid ${{𝒞}{ℐ}}_n$ on a set ${\mathrm{\Omega }}_n$ with $n$ elements, i.e. the inverse submonoid of the symmetric inverse monoid on ${\mathrm{\Omega }}_n$ consisting of all restrictions of the elements of a cyclic subgroup of order $n$ acting cyclically on ${\mathrm{\Omega }}_n$. We show that ${{𝒞}{ℐ}}_n$ has rank $2$ (for $n\ge 2$) and $n{2}^n-n+1$ elements. Moreover, we give presentations of ${{𝒞}{ℐ}}_n$ on $n+1$ generators and $\frac{1}{2}(n^2+3n+4)$ relations and on $2$ generators and $\frac{1}{2}(n^2-n+6)$ relations. We also consider the remarkable inverse submonoid ${{𝒪}{𝒞}{ℐ}}_n$ of ${{𝒞}{ℐ}}_n$ constituted by all its order-preserving transformations. We show that ${{𝒪}{𝒞}{ℐ}}_n$ has rank $n$ and $3\cdot 2^n-2n-1$ elements. Furthermore, we exhibit presentations of ${{𝒪}{𝒞}{ℐ}}_n$ on $n+2$ generators and $\frac{1}{2}(n^2+3n+8)$ relations and on $n$ generators and $\frac{1}{2}(n^2+3n)$ relations.

In this paper, the group homomorphism graph is introduced and investigated. The group homomorphism graph, denoted by $HI(G,G^{\prime })$, is an undirected graph in which the vertex set contains all homomorphisms excluding the monomorphisms and the zero homomorphism from the group $G$ to the group $G^{\prime }$, and two distinct vertices are adjacent if and only if the intersection of their kernels is non-trivial. We investigate the interplay between the graph-theoretic properties of this graph with algebraic properties of the groups. In this work, connectedness, diameter, clique number, chromatic number, domination number, independence number, etc. are found.

In this short report, a brief introduction to Arkani-Hamed, Cohen, Georgi model (ACG-model, (de)constructing dimensions model), whose main characters are that extra-dimensional space-time are generated dynamically from a four-dimensional gauge theory and that extra dimensions are lattices, will be given first. Then after a concise review of NCG on cyclic groups, actions for gauge fields along extra dimensions will be constructed by virtue of NCG and classical (vacuum) solutions will be solved, with low energy phenomenology being classified accordingly. As a conclusion, the behavior of spontaneous symmetry broken within ACG-model can be determined by noncommutative Yang-Mills theory.