scholarly journals The Energy of Conjugate Graph of Some Finite Metabelian Groups of Order Less Than 24

2021 ◽  
pp. 45-51
Author(s):  
M. S. Mahmud ◽  
A. A. Malle ◽  
C. Chibuisi ◽  
M. Z. Idris
Keyword(s):  
Metabelian Group ◽  
Conjugacy Classes ◽  
Metabelian Groups ◽  
Central Elements ◽  
Programming Software

The conjugacy classes of the Metabelian group G, plays an important role in defining the conjugate graph, whose vertices are non-central elements of G, and two vertices are connected if and only if they are conjugate. The constructions of conjugate graphs of all non abelian metabelian groups of order less than 24 are the basis for this paper. And the obtained results are then used to calculate the energy of the aforementioned group. This is aided by specialized programming software (maple).

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.

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.

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.   

Groupings of Metabelian Groups and Extension Categories

1980 ◽  
Vol 32 (2) ◽  
pp. 449-459 ◽  
Author(s):  
K. W. Roggenkamp
Keyword(s):  
Group Ring ◽  
General Result ◽  
Prime Divisor ◽  
Characteristic Zero ◽  
Integral Domain ◽  
Metabelian Group ◽  
Metabelian Groups ◽  
Exact Sequences

Let G be a metabelian group and R an integral domain of characteristic zero, such that no rational prime divisor of │G│ is invertible in R. By RG we denote the group ring of G over R. In this note we shall proveTHEOREM. If RG ≌ RH as R-algebras, then G ≌ HThe question whether this result holds was posed to me by S. K. Sehgal. The result for R = Z is contained in G. Higman's thesis, and he apparently also proved a more general result. At any rate, I think that the methods of the proof are interesting eo ipso, since they establish a “Noether-Deuring theorem” for extension categories.In proving the above result, it is necessary to study closely the category of extensions (ℊs, S), where the objects are short exact sequences of SG-modules

scholarly journals Metabelian groups with the same finite quotients

1974 ◽  
Vol 11 (1) ◽  
pp. 115-120 ◽  
Author(s):  
P.F. Pickel
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Metabelian Group ◽  
Metabelian Groups ◽  
Isomorphism Classes ◽  
Split Extensions ◽  
Finite Quotients ◽  
Finitely Presented

Let F(G) denote the set of isomorphism classes of finite quotients of the group G. Two groups G and H are said to have the same finite quotients if F(G) = F(H). We construct infinitely many nonisomorphic finitely presented metabelian groups with the same finite quotients, using modules over a suitably chosen ring. These groups also give an example of infinitely many nonisomorphic split extensions of a fixed finitely presented metabelian group by a fixed finite abelian group, all having the same finite quotients.

On metabelian groups of prime-power exponent

1969 ◽  
Vol 310 (1502) ◽  
pp. 393-399 ◽  
Keyword(s):  
Prime Power ◽  
Metabelian Group ◽  
Power Exponent ◽  
Metabelian Groups ◽  
The Right

Gupta, Newman & Tobin (1968) show that in a metabelian group of exponent dividing p k , the subgroup generated by p k -1 th powers is nilpotent. In this paper we obtain the ‘right’ bound for the class of this subgroup together with some information about the subgroup generated by p h th powers, thus answering a question raised by Gupta et al .

A note on the cohomology of metabelian groups

1985 ◽  
Vol 98 (3) ◽  
pp. 437-445 ◽  
Author(s):  
P. H. Kropholler
Keyword(s):  
Finite Rank ◽  
Metabelian Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Soluble Groups

The cohomology of finitely generated metabelian groups has been studied in a series of papers by Bieri, Groves, and Strebel [2, 3, 4]. In particular, Bieri and Groves [2] have shown that every metabelian group of type (FP)∞ is of finite rank, and so is virtually of type (FP). The purpose of the present paper is to provide a generalization of this result and to use our methods to prove the existence of a pathological class of finitely generated soluble groups.

Dehn functions of finitely presented metabelian groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wenhao Wang
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Metabelian Group ◽  
Finitely Generated ◽  
Dehn Function ◽  
Metabelian Groups ◽  
Dehn Functions ◽  
Finitely Presented

Abstract In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.

∀-free metabelian groups

1997 ◽  
Vol 62 (1) ◽  
pp. 159-174 ◽  
Author(s):  
Olivier Chapuis
Keyword(s):  
Free Group ◽  
Metabelian Group ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Explicit Description ◽  
Universal Theory ◽  
Metabelian Groups ◽  
Free Metabelian Group ◽  
Finitely Generated Groups

In 1965, during the first All-Union Symposium on Group Theory, Kargapolov presented the following two problems: (a) describe the universal theory of free nilpotent groups of class m; (b) describe the universal theory of free groups (see [18, 1.28 and 1.27]). The first of these problems is still open and it is known [25] that a positive solution of this problem for an m ≤ 2 should imply the decidability of the universal theory of the field of the rationals (this last problem is equivalent to Hilbert's tenth problem for the field of the rationals which is a difficult open problem; see [17] and [20] for discussions on this problem). Regarding the second problem, Makanin proved in 1985 that a free group has a decidable universal theory (see [15] for stronger results), however, the problem of deriving an explicit description of the universal theory of free groups is open. To try to solve this problem Remeslennikov gave different characterization of finitely generated groups with the same universal theory as a noncyclic free group (see [21] and [22] and also [11]). Recently, the author proved in [8] that a free metabelian group has a decidable universal theory, but the proof of [8] does not give an explicit description of the universal theory of free metabelian groups.

The free centre-by-metabelian groups

1973 ◽  
Vol 16 (3) ◽  
pp. 294-299 ◽  
Author(s):  
Chander Kanta Gupta
Keyword(s):  
Characteristic Zero ◽  
Abelian Subgroup ◽  
Integral Domain ◽  
Metabelian Group ◽  
Invariant Subgroup ◽  
Elementary Abelian Subgroup ◽  
Finitely Generated ◽  
Metabelian Groups

Let be the free centre-by-metabelian group of rank n. In this paper, our main result is the followingTheorem. For n ≧ 4, Gn has a finite elementary abelian subgroup Hn of rank . More precisely, Hn is a minimal fully invariant subgroup contained in the centre of Gn and Gn/Hn is isomorphic to a group of 3 x 3 matrices over a finitely generated integral domain of characteristic zero.

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