Bounds for the relative n-th nilpotency degree in compact groups

The line of investigation of the present paper goes back to a classical work of W. H. Gustafson of the 1973, in which it is described the probability that two randomly chosen group elements commute. In the same work, he gave some bounds for this kind of probability, providing information on the group structure. We have recently obtained some generalizations of his results for finite groups. Here we improve them in the context of the compact groups.Comment: 9 pages; to appear in Asian-European Journal of Mathematics with several improvement

Some considerations on the n-th commutativity degrees of finite groups

Let G be a finite group and n a positive integer. The n-th commutativity degree P-n(G) of G is the probability that the n-th power of a random element of G commutes with another random element of G. In 1968, P. Erdos and P.Turan investigated the case n = 1, involving only methods of combinatorics. Later several authors improved their studies and there is a growing literature on the topic in the last 10 years. We introduce the relative n-th commutativity degree P-n(H, G) of a subgroup H of G. This is the probability that an n-th power of a random element in H commutes with an element in G. The influence of P, (G) and P-n (H, G) on the structure of G is the purpose of the present work

Energy of Cayley graphs for alternating groups

Let G be a finite group and S be a subset of G where S does not include the identity of G and is inverse closed. A Cayley graph of a finite group G with respect to the subset S is a graph where the vertices are the elements of G and two vertices a and b in G are adjacent if ab−1 are in the set S. For a simple graph, the energy of a graph can be determined by its eigenvalues. Let Γ be a simple graph. Then by the summation of the absolute values of the eigenvalues of the adjacency matrix of the graph, its energy can be determined. This paper presents the Cayley graphs of alternating groups with respect to the subset S of valency 1 and 2. From the Cayley graphs, the eigenvalues are computed by using some properties of special graphs and then used to compute their energy

Unitarily invariant norm inequalities for operators

We present several operator and norm inequalities for Hilbert space operators. In particular, we prove that if $A_{1},A_{2},...,A_{n}\in {\mathbb B}({\mathscr H})$, then $$|||A_{1}A_{2}^{*}+A_{2}A_{3}^{*}+...+A_{n}A_{1}^{*}|||\leq|||\sum_{i=1}^{n}A_{i}A_{i}^{*}|||,$$ for all unitarily invariant norms. We also show that if $A_{1},A_{2},A_{3},A_{4}$ are projections in ${\mathbb B}({\mathscr H})$, then &&|||(\sum_{i=1}^{4}(-1)^{i+1}A_{i})\oplus0\oplus0\oplus0|||&\leq&|||(A_{1}+|A_{3}A_{1}|)\oplus (A_{2}+|A_{4}A_{2}|)\oplus(A_{3}+|A_{1}A_{3}|)\oplus(A_{4}+|A_{2}A_{4}|)||| for any unitarily invariant norm.Comment: 10 pages, Accepted pape

The topological indices of the non-commuting graph for symmetric groups

Topological indices are the numerical values that can be calculated from a graph and it is calculated based on the molecular graph of a chemical compound. It is often used in chemistry to analyse the physical properties of the molecule which can be represented as a graph with a set of vertices and edges. Meanwhile, the non-commuting graph is the graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if they do not commute. The symmetric group, denoted as, is a set of all permutation under composition. In this paper, two of the topological indices, namely the Wiener index and the Zagreb index of the non-commuting graph for symmetric groups of order 6 and 24 are determined

The Harary index of the non-commuting graph for dihedral groups

Assume G is a non-abelian group which consists a set of vertices, V = {v1 , v2, ..., vn} and a set of edges, E = {e1, e2, ..., em} where n and m are the positive integers. The non-commuting graph of G, denoted by ΓG, is the graph of vertex set G−Z(G), whose vertices are non-central elements, in which Z(G) is the center of G and two distinct vertices are adjacent if and only if they do not commute. In addition, the Harary index of a graph ΓG is the half-sum of the elements in the reciprocal distance of Dij where Dij the distance between vertex i and vertex j. In this paper, the Harary index of the non-commuting graph for dihedral groups is determined and its general formula is developed

Parametric architecture in it’s second phase of evolution

This paper seeks to illustrate the evolution history of Parametric Architecture and describe the reasons why parametric architecture, in its second phase of evolution, called “Parametricism 2.0”, is showing promising abilities in solving more and more intricate socio-environmental problems. In this sense, paper discusses that mentioned school of architecture can be used in several fields other than mere form finding and geometrical coding. Current article studies the history of parametric architecture by finding the root of its name, reviewing its early designs and discussing the work of two of its precursors; then moves on to examine the current situation of the style and defines the word: Parametricism. Paper continues to study the vistas ahead by presenting techniques that empower Parametricism and concludes its discussion by presenting a redefinition for Parametricism. Overall, the paper depicts how “Parametricism 2.0” intends to go back to solving socio-environmental problems; Problems that all the existing evolutionary and generative techniques were initially designed in order to answer them

The exterior degree of a pair of finite groups

The exterior degree of a pair of finite groups $(G,N)$, which is a generalization of the exterior degree of finite groups, is the probability for two elements $(g,n)$ in $(G,N)$ such that $g\wedge n=1$. In the present paper, we state some relations between this concept and the relative commutatively degree, capability and the Schur multiplier of a pair of groups.Comment: To appear in Mediterr. J. Mat