ABSTRACT
This dissertation presents a studyof a certain algebraic system,termed: “Non-commutative
general linear rhotrix group”. The groupis considered to be analogous to the well-known
General Linear Group. The Non-commutative General Linear Rhotrix Group consists of
all invertible rhotrices of size with entries from an arbitrary field F and it has
beenshown to possessnon-commutative rhotrix groups as its subgroups. Certain
subgroups of non-commutative general linear rhotrix group have also beenidentified and
then shown to be embedded in some subgroups of the general linear group. Furthermore,
some finite non-commutative groups of rhotrices as well as their subgroups
areconstructed and shown as concrete examples. It is of interest that this study will go to
a large extent in simplification of teaching and learning of group theory in Mathematical
discipline.
TABLE OF CONTENTS
Cover page……………………………………………………………………… i
Fly leaf………………………………………………………………………….. ii
Title page……………………………………………………………………….. iii
Declaration……………………………………………………………………… iv
Certification…………………………………………………………………….. v
Dedication……………………………………………………………………….. vi
Acknowledgement………………………………………………………………. vii
Abstract………………………………………………………………………….. x
Table of contents………………………………………………………………… xi
List of notations and symbols…………………………………………………… xiii
1.0 CHAPTER ONE : GENERAL INTRODUCTION 1
1.1 Introduction…………..………………………………………………….. 1
1.2 Background of the Study……………………………………………….. 1
1.3 Aim and Objectives of the Study……………………………………….. 4
1.4 Justification of Study…………………………………………………….. 5
1.5 Research Methodology………………………………………………….. 6
1.6 Scope and Limitation …………………………………………………… 6
1.7 Definition of Terms……………………………………………………… 7
1.8 Outline of the dissertation……………………………………………… 13
2.0 CHAPTER TWO :LITERATURE REVIEW 14
2.1 Rhotrix Theory……………….………………………………………….. 14
xi
2.2Commutative rhotrix theory…………………………………………….. 16
2.3 Non-Commutative rhotrix theory……………………………………….. 27
2.4 General Linear Group..………………………………………………….. 41
3.0 CHAPTER THREE : THE NON-COMMUTATIVE RHOTRIX
GROUPS 44
3.1 Introduction…………………………………………………………….. 44
3.2 The Non-Commutative General Linear Rhotrix Group…………….. 44
3.3 Subgroups of Non-commutative General Linear Rhotrix Group…… 52
3.4 Isomorphism between Subgroups of Non-commutative General
LinearRhotrix Group…………………………………………………………..
74
4.0 CHAPTER FOUR : CONSTRUCTION OF FINITE
NON-COMMUTATIVERHOTRIX GROUPS 77
4.1 Introduction.……………………………………………………………… 77
4.2 The Non-CommutativeRhotrix Group over a Finite Field …………… 77
4.3 Finite RhotrixGroup of Size 3 over 2 Z and its Subgroups….……….. 78
4.4 Finite Rhotrix Groups of Size 3 over 3 Z and its Subgroups………….. 84
xii
5.0 CHAPTER FIVE : SUMMARY, CONCLUSION AND RECOMMENDATIONS 103
5.1 Summary ………………………………………………………………… 103
5.2 Conclusion …………………….………………………………………… 103
5.3 Recommendations …………………………………………………………. 104
REFERENCES…………………………………………………… …….
CHAPTER ONE
GENERAL INTRODUCTION
1.1 Introduction
This chapter presents the general introduction of the dissertation, showing the
background of the research, aim and objectives of the study, methodology for carrying
out the research, definition of termsand outline of the dissertation.
1.2 Background of Research
Mathematics is significant and indispensable for any individual regardless of culture or
age. Mathematics provides a powerful universal language and intellectual toolkit for
abstraction, generalization and synthesis. It is the language of science and technology
which enables us to probe the natural universe and develop new technologies that have
helped us control and master our environment, change societal expectations and standards
of living. Mathematical skills are highly valued and sought after. Mathematical training
disciplines the mind, develops logical and critical reasoning, and develops analytical and
problemsolving skills to a high degree. Smith (2004)
Rhotrix theory is a relatively new area of mathematical discipline which deals with
algebra and analysis of array of numbers in mathematical rhomboid form. It beganwith
the work of Ajibade (2003), wherethe concept, algebra, and analysis of rhotrices which
was motivated by the ideas on matrix-tersions and matrix-noitrets proposed by Atanassov
and Shannon (1998). Ajibade (2003)gave the initial definition of rhotrix of size 3 as a
2
mathematical array that is in some way, between two-dimensional vectors and 2 2
dimensional matrices. Since then, many authors (Mohammed (2009), Tudunkaya etal
(2010b), Aminu (2010c), Mohammed etal (2011), Mohammed etal (2012), etc.) have
shown interest in the usage of rhotrix set, as an underlying set, for construction of
algebraic structures.
Sani (2004) proposed an alternative method for multiplication of rhotrices of size three,
based on their rows and columns, as comparable to matrix multiplication. This was in an
attempt to answer the question of „how can we transform a rhotrix to a matrix and vice
versa‟posed by Ajibade (2003) in the concluding remarks of his article. This method of
rhotrix multiplication is now referred to as „row-column-based method for rhotrix
multiplication‟. Unlike Ajibade‟s method of multiplication, that is both commutative and
associative, Sani‟s method of rhotrix multiplication is non-commutative but
associative.The alternative method for multiplication of base rhotrices proposed by Sani
(2004) was later generalized to include rhotrices ofsize n in Sani (2007).
It was also shown in Sani (2007) that there exists an isomorphic relationship between the
group of all invertible rhotrices of size nand the group of all invertible w wdimensional
matrices, where
2
n 1
w and n 2Z 1.
Thus, two methods for multiplication of rhotrices are presently available in the literature
of rhotrix theory. In this work, we shall refer to the method for multiplication of rhotrices
defined in Ajibade (2003) as “commutative method for rhotrix multiplication” and the
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alternative method for multiplication of rhotrices defined in Sani (2004, 2007) as “noncommutative
method for rhotrix multiplication”.
Mohammed (2007a) adopted the commutative method for rhotrix multiplication to
propose classification of rhotrices and their expression as algebraic structures of groups,
semigroups, monoids, rings and Boolean algebras.
Based on non-commutative method for rhotrix multiplication, the search for a
transformation of a rhotrix to a matrix and vice-versa was completely achieved in Sani
(2008), where he proposed a method of converting rhotrix to a special form of matrix
called „coupled matrix’. This coupled matrix was used to solve two different systems of
linear equations simultaneously, where one is an n n system while the other one is an
(n 1) (n 1) system. Following this idea, Sani (2009) presented the solution of two
coupled matrices by extending the idea of a coupled matrix presented in his earlier work
to a general case involving (m n) and (m 1) (n 1) matrices.
It is noteworthy to mention that any research work by interested author(s) in the literature
of rhotrix theory is based on either commutative method or non-commutative method for
rhotrix multiplication. In the presentation of thisstudy on the non-commutative general
linear rhotrix group, it is obvious that we adopt the non-commutative multiplication
technique proposed in Sani (2004, 2007). The reason behind the choice is that an
algebraically non-commutative group offers an exciting platform for carrying out
mathematical research in group theory and number theory. Above all, this area is still
fertile and beckons on researchers to explore the rich properties analogous to matrices in
the study of rhotrices.
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One of the well-known areas of Mathematics is Group theory. In mathematics a group is
defined as a non empty set having a binary operation defined on it and satisfying four
axioms,as follows: closure, associativity, existence of an identity element and existence
of inverses.Lloyd and Frank (2004). A rhotix group is a group having rhotrix set as the
underlying set.
Group theory has been well developed by researchers before the twentieth century.
General linear group has played a vital role in the study of group representation theory.
Many concepts in rhotrix theory were analogous to matrix theory, but the concept of
rhotrices has two different multiplication methods, providing two parallel study areas for
research. This makes Rhotrix Theory a well-deserved area of research.
This research work is dealing with the study of generalized rhotrix groups. We consider a
rhotrix set, ( ) n R F of size n over a field F , together with the binary operation ofnoncommutative
method for rhotrix multiplication proposed by Sani (2007),in order to
construct certain algebraic systems, which we term as „Rhotrix Groups‟. We identify
certain subgroups of these groups using Lagrange‟s theorem as a check. Furthermore, we
identified certain finite groups of rhotrices of size 3. We also showed isomorphic
relationship between certain groups and certain subgroups within our constructions.
1.3Aim and Objectives of the Study
The aim of this dissertation is to present an algebraic study of the development of noncommutative
general linear rhotrix group. In particular, the following are the research
objectives:
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(i) To develop the basic fundamentals necessary for the algebraic study of the concept of „non-commutative general rhotrix group‟ as a new paradigm of science.
(ii) To identify and study the properties of General Linear Rhotrix Group as analogous to the well-known General Linear Group in the literature.
(iii) To dissect the General Rhotrix Group in order to uncover its subgroups.
(iv) To establish the embedment of a particular subgroup of General Linear Rhotrix Group in a particular subgroup of General Linear Group.
(v) To construct some finite non-commutative groups of rhotrices and identify their subgroups.
1.4 Justification of Research
Since its inception, so much has been done on the algebra of rhotrices but, from existing literature available to us, nothing has been done to advance group theory by using rhotrices as an underlying set for construction of groups. We intend to fill this gap by presenting an algebraic study of non-commutative general linear rhotrix group.We shall also consider finite non-commutative rhotrix groups with entries taken from the set of integers modulo p .
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1.5Research Methodology
The method adopted in this dissertation is by consulting all necessary and relevant papers in the literature on fundamentals of Rhotrix theory, Matrix theory and Group theory in order to obtain background information for developing the theory of non-commutative groups of rhotrices.
In line with the work of Mohammed and Balarabe (2014) on the review of developments in rhotrix theory, an algebraic study of non-commutative general rhotrix group will be presented as an extension to the development of non-commutative rhotrix theory. This will be achieved through our adoption of row-column-based method for rhotrix multiplication defined in Sani (2004, 2007) as the group binary operation.
Next, a dissection of non-commutative general rhotrix group would be made, so as to uncover its subgroups and then establish an isomorphic relationshipbetween them. Furthermore, an introduction of the study of finite groups of rhotrices with entries from the field of integer modulo p, will be considered.
1.6 Scope and Limitation
This study will be limited to the Algebraic properties of non-commutative general linear rhotrix group. The Analysis, Topology and Representation of rhotrix group will not be considered.
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1.7 DEFINITION OF TERMS
The following definitions will be useful in carrying out the work:
Definition 1.7.1 Matrix set
A matrix set ( ) m n M C is a collection of rectangular arrays, called m n dimensional
matrices with entries from the field of complex numbers. Thus,
11 12 13 1
21 22 23 2
11 12
1
… … … … …
… … … … …
: : : : : : : : :
: : : : : : : : :
( ) : : : : : : : : : : , ,…, ,…
: : : : : : : : :
: : : : : : : : :
: : : : : : : : :
… … … … … … …
n
n
m n mn
m mn
a a a a
a a a a
M C a a a C
a a
(1.1)
Definition 1.7.2 Matrix-tertion and matrix-noitret
Matrix-tertion and Matrix-noitret can be defined as mathematical arrays that are in some
way between 2-dimensional vectors and 2 2 -dimensional matrices introduced by
Atanassov and Shannon in (1998). Matrix-tertion and Matrix-noitret are denoted by T
and N respectively and defined as
: , , (1.2)
a b
T a b c C
c
and
8
: , ,
a
N a b c C
b c
(1.3)
Definition 1.7.3 Rhotrix
A rhotrix R of size n is a rhomboidal array with entries from a field F which can be
expressed as a couple of two square matrices A and C of sizes (t t) and (t 1) (t 1) ,
where
1
and 2 1.
2
n
t n Z This can be represented according to Sani (2008) as :
11
21 11 12 11 12 1( 1) 1
21 22 2( 1) 2
( 1)( 1) 1 1
( 1)1 ( 1)2 ( 1)( 1) ( 1)
( 1) ( 1)( 1) ( 1)
…
… …. … … … …
, … … … … … … … … … …
… … … … … …
t t
t t
n t t t t t t
t t t t t
t t t t t t
tt
a
a c a a a a a
a a a a
R A C a a
a a a a
a c a
a
11 1( 1)
( 1)1 ( 1)( 1)
1 2 ( 1)
…
, … … … (1.4)
…
…
t
t t t t
t t t t tt
c c
c c
a a a a
Examples: a rhotix of size 5 can be written as:
5 , (1.5)
a
b c d a d j
c h
R e f g h j b g m
f l
k l m e k n
n
Also, a rhotrix of size 7 can be writtenas
7 , (1.6)
a
b c d
a d j w
e f g h j c h r
b g q x
R k l m p q r w f p v
e m u
s t u v x l t
k s y n
y
n
9
Definition 1.7.4 A set of all rhotrices of size n
The set of all rhotrices of size n with entries from a field F is a collection of all rhotrices
of size n, defined as:
Definition 1.7.5 Determinant of a rhotrix of size n
11
21 11 12
1 1
( 1) ( 1)( 1) ( 1)
… …. … … …
, … … … … …
… … … … …
n ij lk t t
t t t t t t
tt
a
a c a
If R a c a a
a c a
a
,
then the determinant of n R written as 1 det( ) det( Rn At )det(Ct ) where At and t 1 C are the
two square matrices of dimension (t t) and (t 1) (t 1) respectively which make up
the rhotrix n R with
1
2 1.
2
n
t and n Z
11
21 11 12
1 1
( 1) ( 1)( 1) ( 1)
… …. … … …
… … … … …
: ,
( ) … … … … … , (1.7)
1
1 , , 1 , 1; 2 1
2
t t
ij lk
n
t t t t t t
tt
a
a c a
a a
a F c F
R F
a c a
a
n
where i j t l k t t and n Z
10
Definition 1.7.6 The inverse of a rhotrix of size n
The inverse of a rhotrix , n ij lk R a c , is the rhotrix 1 , n ij lk R q r such that
1 , , , n n R R aij clk qij rlk Iij Ilk where ij t t
q and
ij t 1 t 1
r are the inverses of the
two square matrices ij t t
a and
ij t 1 t 1
c respectively,which make up the rhotrix n R with
1
2 1.
2
n
t and n Z
Remark: A rhotrix n R is said to be invertible or non- singular if the determinant is nonzero.
That is n R is invertible iff det( ) 0 n R .
Definition 1.7.7 A set of all invertible rhotrices of size n
A collection of all invertible rhotrices of size with entries from an arbitrary field F ,
denoted by ( ) n GR F is called a set of all invertible rhotrices of sizen.
Definition 1.7.8Operations of addition and multiplication on rhotrices of size n
Let n A and n B be two rhotrices of size n. The operation of addition of n A and n B is
defined according to Mohammed (2011) as:
11
11 11
21 11 12 21 11 12
1 1 1 1
( 1) ( 1)( 1) ( 1) ( 1) ( 1)( 1) ( 1)
… … … … … …
… … … … … … … … … …
… … … … … …
n n t t t t
t t t t t t t t t t t t
tt tt
p u
p q p u v u
A B p p u u
p q p u v u
p u
( 1)
11 11
21 21 11 11 12 12
1 1 1 1
( 1) ( 1) ( 1)( 1) ( 1)( 1) ( 1)
… … …
… … … … … (1.8)
… … …
t t
t t t t
t t t t t t t t t t
tt tt
p u
p u q v p u
p u p u
p u p v p u
p u
and the operation of multiplication of n A and n B is defined according to Sani (2008)
as
1 1 1 1 2 2 2 2
An Bn pi j ,ql k ui j ,vl k
1 1 2 2 1 1 2 2
2 1 2 1
1
1 1
( ), ( ) (1.9)
t t
i j i j l k l k
i j l k
p u q v
Definition 1.7.9 Group
A group ( ( ), ) n GR F is a non-empty set ( ) n GR F together with a binary operation ( )
satisfying the following properties:
( ( ), )1 n GR F : Closure: for all , ( ( ), ) n x y GR F , ( ( ), ) n x y GR F .
( ( ), )2 n GR F : Associativity: for all , , ( ( ), ) n x y z GR F , (x y) z x (y z) .
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( ( ), )3 n GR F : Existence of identity: ( ( ), ) : ( ( ), ) n n e GR F x e e x x GR F
( ( ), )4 n GR F :Existence of inverse: each element of ( ( ), ) n GR F possesses an inverse.
That is,
1 1 1 ( ( ), ) : , ( ( ), ) n n x GR F x x x x e x GR F
Definition 1.7.10 Subgroup
A non-empty subset ( ( ), ) n H F of a group ( ( ), ) n GR F is called a subgroup of ( ( ), ) n GR F if
( ( ), ) n H F is a group under the binary operation ‘ ‘defined on ( ( ), ) n GR F .
Definition 1.7.11 Rhotrix group
A rhotrix group is a group having rhotrix set as an underlying set.
Definition 1.7.12 Rhotrix subgroup
A non-empty subset ( ) n H F of a rhotrix group ( ( ), ) n GR F is said to be a rhotrix subgroup of
( ( ), ) n GR F if the composition in ( ( ), ) n GR F is also a composition in ( ( ), ) n H F and for this
composition, ( ( ), ) n H F is itself a rhotrix group.
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1.8 OUTLINE OF THE DISSERTATION
The outline of the dissertationis as follows:
Chapter one presents the general introduction of the dissertation, the aim and objectives
of the study, the methodology for carrying out the research and definition of terms.
Chapter two focuses on a review of developments in the literature of rhotrix theory,
starting from the year 2003, when the concept of rhotrix was introduced up to the end of
2014.
Chapterthree considers the rhotrix set ( ) n GR F consisting of all invertible rhotrices of
sizenwith entries from a given field F and together with row-column-based method for
rhotrix multiplication, in order toinitiate the concept of non-commutative general linear
rhotrix group. We identify and characterize its subgroups. Furthermore, we investigate
some relationships between the non-commutative general linear rhotrix group and the
well-known general linear group.
Chapter four introduces concrete constructions of finite non-commutative groups of base
rhotrices. Ideas in the study of permutations and number theory are employed to ascertain
the number of elements in each group, bearing in mind the axiomatic requirements for a
rhotrix group.
Chapter five gives the summary, conclusion and recommendations for future researches.
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