scholarly journals On Some Graphs of Finite Metabelian Groups of Order Less Than 24

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 237-247
Author(s):  
Ibrahim Gambo ◽  
Nor Haniza Sarmin ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Normal Subgroup ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Metabelian Group ◽  
Factor Group ◽  
Conjugacy Classes ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup ◽  
Class Graph

In this work, a non-abelian metabelian group is represented by G while represents conjugacy class graph. Conjugacy class graph of a group is that graph associated with the conjugacy classes of the group. Its vertices are the non-central conjugacy classes of the group, and two distinct vertices are joined by an edge if their cardinalities are not coprime. A group is referred to as metabelian if there exits an abelian normal subgroup in which the factor group is also abelian. It has been proven earlier that 25 non-abelian metabelian groups which have order less than 24, which are considered in this work, exist. In this article, the conjugacy class graphs of non-abelian metabelian groups of order less than 24 are determined as well as examples of some finite groups associated to other graphs are given.

TTHE CONJUGACY CLASSES OF METABELIAN GROUPS OF ORDER AT MOST 24

2015 ◽  
Vol 77 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Ibrahim Gambo ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Normal Subgroup ◽  
Conjugacy Class ◽  
Equivalence Relation ◽  
Metabelian Group ◽  
Equivalence Classes ◽  
Factor Group ◽  
Conjugacy Classes ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup

In this paper, G denotes a non-abelian metabelian group and denotes conjugacy class of the element x in G. Conjugacy class is an equivalence relation and it partitions the group into disjoint equivalence classes or sets. Meanwhile, a group is called metabelian if it has an abelian normal subgroup in which the factor group is also abelian. It has been proven by an earlier researcher that there are 25 non-abelian metabelian groups of order less than 24 which are considered in this paper. In this study, the number of conjugacy classes of non-abelian metabelian groups of order less than 24 is computed.

THE CONJUGACY CLASSES OF SOME FINITE METABELIAN GROUPS AND THEIR RELATED GRAPHS

2016 ◽  
Vol 79 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Ain Asyikin Ibrahim ◽  
Alia Husna Mohd Noor ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Graph Theory ◽  
Conjugacy Class ◽  
Dihedral Group ◽  
Clique Number ◽  
Conjugacy Classes ◽  
Quaternion Group ◽  
Metabelian Groups ◽  
Dominating Number ◽  
Graph Properties ◽  
Class Graph

In this paper, the conjugacy classes of three metabelian groups, namely the Quasi-dihedral group, Dihedral group and Quaternion group of order 16 are computed. The obtained results are then applied to graph theory, more precisely to conjugate graph and conjugacy class graph. Some graph properties such as chromatic number, clique number, dominating number and independent number are found.   

Degrees of faithful irreducible representations of metabelian groups

2021 ◽  
pp. 2250181
Author(s):  
Rahul Dattatraya Kitture ◽  
Soham Swadhin Pradhan
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Positive Characteristic ◽  
Number Fields ◽  
Irreducible Representations ◽  
Modular Representations ◽  
Metacyclic Group ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup ◽  
Field Of Positive Characteristic

In 1993, Sim proved that all the faithful irreducible representations of a finite metacyclic group over any field of positive characteristic have the same degree. In this paper, we restrict our attention to non-modular representations and generalize this result for — (1) finite metabelian groups, over fields of positive characteristic coprime to the order of groups, and (2) finite groups having a cyclic quotient by an abelian normal subgroup, over number fields.

scholarly journals General Form of Domination Polynomial for Two Types of Graphs Associated to Dihedral Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 149-155
Author(s):  
Nabilah Najmuddin ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Complete Graphs ◽  
Dominating Sets ◽  
Dihedral Groups ◽  
Graph Polynomial ◽  
Central Elements ◽  
Domination Polynomial ◽  
Class Graph

A domination polynomial is a type of graph polynomial in which its coefficients represent the number of dominating sets in the graph. There are many researches being done on the domination polynomial of some common types of graphs but not yet for graphs associated to finite groups. Two types of graphs associated to finite groups are the conjugate graph and the conjugacy class graph. A graph of a group G is called a conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate to each other. Meanwhile, a conjugacy class graph of a group G is a graph in which its vertices are the non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The conjugate and conjugacy class graph of dihedral groups can be expressed generally as a union of complete graphs on some vertices. In this paper, the domination polynomials are computed for the conjugate and conjugacy class graphs of the dihedral groups.

scholarly journals NORMAL SUBGROUPS WHOSE CONJUGACY CLASS GRAPH HAS DIAMETER THREE

2016 ◽  
Vol 94 (2) ◽  
pp. 266-272
Author(s):  
ANTONIO BELTRÁN ◽  
MARÍA JOSÉ FELIPE ◽  
CARMEN MELCHOR
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
A Finite Group ◽  
Class Graph

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We determine the structure of $N$ when the diameter of the graph associated to the $G$-conjugacy classes contained in $N$ is as large as possible, that is, equal to three.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

scholarly journals The Laplacian Energy of Conjugacy Class Graph of Some Finite Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (1) ◽  
pp. 59-65
Author(s):  
Rabiha Mahmoud ◽  
Amira Fadina Ahmad Fadzil ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Laplacian Eigenvalues ◽  
Laplacian Matrices ◽  
Laplacian Energy ◽  
The Difference ◽  
Generalized Quaternion ◽  
Class Graph

Let G be a dihedral group and its conjugacy class graph. The Laplacian energy of the graph, is defined as the sum of the absolute values of the difference between the Laplacian eigenvalues and the ratio of twice the edges number divided by the vertices number. In this research, the Laplacian matrices of the conjugacy class graph of some dihedral groups, generalized quaternion groups, quasidihedral groups and their eigenvalues are first computed. Then, the Laplacian energy of the graphs are determined.

Finite Groups with a Disconnectedp-Regular Conjugacy Class Graph

2004 ◽  
Vol 32 (9) ◽  
pp. 3503-3516 ◽  
Author(s):  
Antonio Beltrán ◽  
María José Felipe
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Class Graph

scholarly journals The Energy of Conjugacy Classes Graphs of Some Order of Alternating Groups

2020 ◽  
Vol 7 (4) ◽  
pp. 62-71
Author(s):  
Zuzan Naaman Hassan ◽  
Nihad Titan Sarhan
Keyword(s):  
Conjugacy Class ◽  
Adjacency Matrix ◽  
Finite Graph ◽  
Conjugacy Classes ◽  
Alternating Group ◽  
Alternating Groups ◽  
Eigen Values ◽  
Finite Set ◽  
Class Graph ◽  
Square Matrix

The energy of a graph , is the sum of all absolute values of the eigen values of the adjacency matrix which is indicated by . An adjacency matrix is a square matrix used to represent of finite graph where the rows and columns consist of 0 or 1-entry depending on the adjacency of the vertices of the graph. The group of even permutations of a finite set is known as an alternating group  . The conjugacy class graph is a graph whose vertices are non-central conjugacy classes of a group , where two vertices are connected if their cardinalities are not coprime. In this paper, the conjugacy class of alternating group  of some order for   and their energy are computed. The Maple2019 software and Groups, Algorithms, and Programming (GAP) are assisted for computations.

ON -PARTS OF CONJUGACY CLASS SIZES OF FINITE GROUPS

2018 ◽  
Vol 97 (3) ◽  
pp. 406-411 ◽  
Author(s):  
YONG YANG ◽  
GUOHUA QIAN
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let $G$ be a finite group. Let $\operatorname{cl}(G)$ be the set of conjugacy classes of $G$ and let $\operatorname{ecl}_{p}(G)$ be the largest integer such that $p^{\operatorname{ecl}_{p}(G)}$ divides $|C|$ for some $C\in \operatorname{cl}(G)$. We prove the following results. If $\operatorname{ecl}_{p}(G)=1$, then $|G:F(G)|_{p}\leq p^{4}$ if $p\geq 3$. Moreover, if $G$ is solvable, then $|G:F(G)|_{p}\leq p^{2}$.

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