scholarly journals Commuting Graphs, C(G, X) in Symmetric Groups Sym(n) and Its Connectivity

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1178
Author(s):  
Athirah Nawawi ◽  
Sharifah Kartini Said Husain ◽  
Muhammad Rezal Kamel Ariffin
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Commutativity Property

A commuting graph is a graph denoted by C ( G , X ) where G is any group and X, a subset of a group G, is a set of vertices for C ( G , X ) . Two distinct vertices, x , y ∈ X , will be connected by an edge if the commutativity property is satisfied or x y = y x . This study presents results for the connectivity of C ( G , X ) when G is a symmetric group of degree n, Sym ( n ) , and X is a conjugacy class of elements of order three in G.

scholarly journals A Note on Commuting Graphs for Symmetric Groups

10.37236/95 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
C. Bates ◽  
D. Bundy ◽  
S. Hart ◽  
P. Rowley
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Vertex Set

The commuting graph ${\cal C}(G,X)$, where $G$ is a group and $X$ a subset of $G$, has $X$ as its vertex set with two distinct elements of $X$ joined by an edge when they commute in $G$. Here the diameter and disc structure of ${\cal C}(G,X)$ is investigated when $G$ is the symmetric group and $X$ a conjugacy class of $G$.

scholarly journals On Commuting Graphs for Elements of Order 3 in Symmetric Groups

10.37236/2362 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Athirah Nawawi ◽  
Peter Rowley
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Vertex Set

The commuting graph $\mathcal{C}(G,X)$, where $G$ is a group and $X$ is a subset of $G$, is the graph with vertex set $X$ and distinct vertices being joined by an edge whenever they commute. Here the diameter of $\mathcal{C}(G,X)$ is studied when $G$ is a symmetric group and $X$ a conjugacy class of elements of order $3$.

scholarly journals The Topological Indices of the Non-commuting Graph for Symmetric Groups

2020 ◽  
pp. 1-5
Author(s):  
Nur Idayu Alimon ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Symmetric Group ◽  
Chemical Compound ◽  
Molecular Graph ◽  
Symmetric Groups ◽  
Wiener Index ◽  
Topological Indices ◽  
Zagreb Index ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set

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 S_n, 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. Keywords: Wiener index; Zagreb index; non-commuting graph; symmetric groups

scholarly journals New characterization of symmetric groups of prime degree

2017 ◽  
Vol 9 (1) ◽  
pp. 5-12
Author(s):  
A. K. Asboei ◽  
R. Mohammadyari
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Class Size ◽  
Symmetric Groups ◽  
Prime Degree ◽  
Conjugacy Class Size

Abstract In this paper, will show that a symmetric group of prime degree p ≥ 5 is recognizable by its order and a special conjugacy class size of (p − 1)!.

scholarly journals On diameters estimations of the commuting graphs of Sylow $p$-subgroups of the symmetric groups

2013 ◽  
Vol 5 (1) ◽  
pp. 70-78 ◽  
Author(s):  
Yu.Yu. Leshchenko ◽  
L.V. Zoria
Keyword(s):  
Undirected Graph ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Central Elements

The commuting graph of a group $G$ is an undirected graph whose vertices are non-central elements of $G$ and two distinct vertices $x,y$ are adjacent if and only if $xy=yx$. This article deals with the properties of the commuting graphs of Sylow $p$-subgroups of the symmetric groups. We define conditions of connectedness of respective graphs and give estimations of the diameters if graph is connected.

scholarly journals Certain unitary representations of the infinite symmetric group, II

1987 ◽  
Vol 106 ◽  
pp. 143-162 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Discrete Group ◽  
Inductive Limit ◽  
Symmetric Groups ◽  
Discrete Groups ◽  
Type I ◽  
Infinite Symmetric Group ◽  
Locally Finite ◽  
Distinctive Features ◽  
Infinite Dimensional

The infinite symmetric group is the discrete group of all finite permutations of the set X of all natural numbers. Among discrete groups, it has distinctive features from the viewpoint of representation theory and harmonic analysis. First, it is one of the most typical ICC-groups as well as free groups and known to be a group of non-type I. Secondly, it is a locally finite group, namely, the inductive limit of usual symmetric groups . Furthermore it is contained in infinite dimensional classical groups GL(ξ), O(ξ) and U(ξ) and their representation theories are related each other.

scholarly journals The multistep homology of the simplex and representations of symmetric groups

2019 ◽  
Vol 169 (2) ◽  
pp. 231-253
Author(s):  
MARK WILDON
Keyword(s):  
Symmetric Group ◽  
Symmetric Groups ◽  
Explicit Construction ◽  
Fixed Size ◽  
Simplicial Homology ◽  
Chain Complexes ◽  
Basic Spin ◽  
Characteristic Two

AbstractThe symmetric group on a set acts transitively on the set of its subsets of a fixed size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.

Rational permutation groups containing a full cycle

2013 ◽  
Vol 63 (6) ◽  
Author(s):  
Temha Erkoç ◽  
Utku Yilmaztürk
Keyword(s):  
Symmetric Group ◽  
Proper Subgroup ◽  
Symmetric Groups ◽  
Transitive Permutation Group ◽  
Rational Group ◽  
Prime Degree ◽  
A Finite Group ◽  
Rational Groups ◽  
Complex Characters

AbstractA finite group whose irreducible complex characters are rational valued is called a rational group. Thus, G is a rational group if and only if N G(〈x〉)/C G(〈x〉) ≌ Aut(〈x〉) for every x ∈ G. For example, all symmetric groups and their Sylow 2-subgroups are rational groups. Structure of rational groups have been studied extensively, but the general classification of rational groups has not been able to be done up to now. In this paper, we show that a full symmetric group of prime degree does not have any rational transitive proper subgroup and that a rational doubly transitive permutation group containing a full cycle is the full symmetric group. We also obtain several results related to the study of rational groups.

The normalizer property for integral group rings of holomorphs of finite nilpotent groups and the symmetric groups

2017 ◽  
Vol 16 (02) ◽  
pp. 1750025 ◽  
Author(s):  
Jinke Hai ◽  
Shengbo Ge ◽  
Weiping He
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Nilpotent Group ◽  
Symmetric Groups ◽  
Nilpotent Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
A Finite Group ◽  
The Normalizer Property

Let [Formula: see text] be a finite group and let [Formula: see text] be the holomorph of [Formula: see text]. If [Formula: see text] is a finite nilpotent group or a symmetric group [Formula: see text] of degree [Formula: see text], then the normalizer property holds for [Formula: see text].

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