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ALGEBRAIC STUDY OF RHOTRIXGROUPS

 Format: MS-WORD   Chapters: 1-6

 Pages: 1-84   Attributes: DOWNLOADABLE

 Amount: 3,000

 Sep 17, 2022 |  10:56 pm |  81


ALGEBRAIC STUDY OF RHOTRIXGROUPS

1.1

INTRODUCTION

Mathematics is a very rich subject which serves as an integral part of life as a whole. The  application  of  Mathematics  is  related  to  every  sphere  of  life. Professions like Engineering, Economics, Finance, Medicine Science, Pharmacy, just to mention but a few, cannot do well without Mathematics.


Mathematics has a lot of branches among which is Algebra. In algebra, concepts such as linear algebra and abstract algebra are studied.  Of interest in this study are Groups, which is a branch of abstract algebra. The study is intended to look at recent innovations in this area.

The concept   of   rhotrix   was   introduced   by  Ajibade(2003)  as an  extension  of  ideas  on matrix – tertions  and  matrix- noitrets.  Ajibade   presented   the initial concept of the algebra and analysis of rhotrix and established  some interesting  relationship  between  rhotrices  and   their  hearts. A  RhotrixR of dimension three was defined as R

 


a, b   c d   : a, b, c, d , e, e

 

,

 



-

 



-

 



-



(1.1)

 

where h(R)  =  c,  the  element  at the  perpendicular  intersection of the  two  diagonals  of a  rhotrixR  and it   is  called  the  heart  of  R.Thus


1

 

a

R b   h R d

e

 


,is a  specific  rhotrix. After   rhotrix  theory  was   introduced   by   Ajibade   in   (2003), several  authors developed interest  to  research  in the  area.  Construction of algebraic  structures  by way of using  rhotrix set  as the  underlying  set  became  an  innovation.


Sani  (2004)  proposed  an  alternative  method  of  rhotrix  multiplication of  size three.  The  method  is  row-column  based  and  it  is  non-commutative  but  associative. This  is  unlike  that  of  Ajibade  which  is  both  commutative  and  associative.

It  was shown  in Sani  (2004)  that  there  exists  an  isomorphic  relationship  between the  groups of   all   invertible   rhotrices of   size   three   and   the   group   of   all   22   dimensional matrices.

In this dissertation, constructions of groups of rhotrices having entries in sets of residue classes are presented as algebraic structures. The types of rhotrix groupsconsidered in this dissertation  are commutative in property,and they serve as direct extension to those initiated by Mohammed (2007a).


1.2 

STATEMENT OF THE PROBLEM

In   the   literature,   it   has   been   shown   that invertible   matrices   of   a   fixed dimension  form  a  group. Thus, invertible 22 matrices for instance form a group.  It  wil2 therefore  be  very  interesting  to  see  whether  invertible rhotrices  of  any  fixed  size form a  group.

In   this   dissertation, a construction of algebraic groups using   rhotrix sets as the underlying  sets, with entries in residue  classes  of  modulo  two,  three  and  five  were specifically  used.


1.3 RESEARCH AIMAND OBJECTIVES

The   aim  of  this  research  work  is  to construct  rhotrix  groups  bearing  in  mind  the following  objectives:

i. To construct finite algebraic structures that satisfy the axioms of groups using rhotrixsets as                      underlying sets.


ii. To  systematize ideas in  group  theory  to  rhotrix group theory,using  already known                    concepts such as cosets, order, subgroups,   Lagrange’s   theorem of a  group and  so  on.


1.4

RESEARCH METHODOLOGY

In the first state, a fundamental study of ideas in group and rhotrix theories was carried out. Next, Algebraic structures were examined, having satisfied   the   group axioms; they  were   systematized   to rhotrixgroups structures specifically of unit hearts.










 


 


 


 


 


 


 




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