ABSTRACT
In this dissertation, Algebraicproperties ofgroups ofrhotriceswere constructed and somealready known theories in group theory were verified. Cyclic rhotrixgroup was developed from the cyclic multiplication table presented. Subgroups of the rhotrix groups developed were also identified and the order of the rhotrix groups were discussed .Homomorphisms and Isomorphisms of rhotrix groups were equally presented. Finite algebraic structures that satisfy the axioms of groups were constructed using rhotrix sets as the underlying sets.Ideas in group theory were systematized into rhotrix group theory using already known concepts such as cosets, order, subgroups, Lagrange’s theorem of a group and so on. The entries were derived from residue classes of modulo two,three and five.
TABLE OF CONTENTS
Cover Page – – – – – – – – – i Fly leaf – – – – – – – – – ii Title Page – – – – – – – – – iii Declaration – – – – – – – – – iv Certification – – – – – – – – – v Dedication – – – – – – – – – vi Acknowledgement – – – – – – – – vii Abstract – – – – – – – – – viii Table of Contents – – – – – – – – ix CHAPTER ONE: GENERAL INTRODUCTION 1.1 Introduction – – – – – – – – 1 1.2 Statement of the Problem – – – – – – 2 1.3 Research Aim and Objectives – – – – – 3 1.4 Research Methodology – – – – – – 3 1.5 Significance of the Study – – – – – – 4 1.6 Definition of Terms – – – – – – 4 1.7 Organization of the Dissertation – – – – – – 5 CHAPTER TWO: LITERATURE REVIEW 2.1 Introduction – – – – – – – – 7 2.2 Rhotrix Algebra – – – – – – – 8
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CHAPTER THREE: GROUPS OF RHOTRICES 3.1 Introduction – – – – – – – – 27 3.2 Rhotrix Groups – – – – – – – 27 CHAPTER FOUR:ISOMOPHISM OF RHOTIX GROUPS 4.1 Isomorphic Rhotrix Groups – – – – – – 45 4.2 Multiplication Table for Rhotrix Groups – – – – 52 4.3 CyclicRhotrix Groups – – – – – – 52 CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATIONS 5.1 Summary – – – – – – – – 55 5.2 Conclusion – – – – – – – – 56 5.3 Recommendations – – – – – – – 56 References – – – – – – – – – – 58 Appendix A – – – – – – – – – – – 62 Appendix B – – – – – – – – – 64
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CHAPTER ONE
GENERAL INTRODUCTION
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 andabstract 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 ofrhotrix and established some
interesting relationship between rhotrices and their hearts. A RhotrixR of dimension
three was defined as
: , , , , ,
a
R 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,
2
,
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 22 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 22matrices for instance form a group. It will
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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 rhotrix sets 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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1.5 SIGNIFICANCE OF THE STUDY
The study will reveal the extent to which mathematical enrichment exercises
could be developed to help in extending existing ideas.
The findings of the study will give diversified knowledge on how ideas in
group theory can be extended to the young theory of rhotrices.
1.6 DEFINITIONS OF TERMS
1.6.1 A group: Let G be a non–empty set and a binary operation defined on G. The set G
is said to be a group if the following axioms are satisfied:
a) For all a, b, G, a b G i.e. closure axiom.
b) For all a, b c ∈ 𝐺, 𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 ∗ 𝑏 ∗ 𝑐 i.e. associativity axiom.
c) There exists eG for any aG ae ea a identity axiom.
d) For any 1 1 1 a G a G, a a a a e, that is inverses exist
Further to this, if ab ba then the group is said to be commutative.
1.6.2 A rhotrix group: If a set of rhotriceswith a binary operation satisfy the axioms
of a group, then the set is termed as a rhotrix group.
1.6.3 The symmetric group: Consider the set of permutations of n elements under
the operation of composition of permutations. This set forms a group called the
symmetric group denoted as n S .The order of the group is n-factorial, n! .
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1.6.4 The Cyclic group: If a set has all its elements as powers of a particular
element in the set, then it forms a group called the cyclic group, denoted as n Z if
the elements are integers. In this case if the base element is say a, then all other
elements in the set are n a for some integer n. In particular, the identity element, e,
is m a for some integer m.
1.6.5 HEART OF A RHOTRIX
The element at the perpendicular intersection of the two diagonals of a rhotrixR is
called its heart denoted by ℎ 𝑅 .
1.6.6 EQUALITY OF RHOTRICES
Two rhotricesR and S are said to be equal if both are of the same size and
order, and each element of R is equal to the corresponding element of S for
each pair of the entries.
1.7 ORGANIZATION OF THE DISSERTATION
Apart from chapter one, we give the following details for the remaining chapters.
Chapter two is the literature review, where the past literature on the types of
rhotrices and the operations on rhotrices, such as addition, scalar multiplication and
multiplication 𝑜 , determinant and inverse of a rhotrix were discussed. Also,
concepts discussed by earlier contributors by way of seminars and contributed
papers will be looked into, so as to provide a better and wider view for
understanding.
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In chapter three, various theorems, rules and concepts of group theory in relation to rhotrices are presented. In chapter four, the discussion centered on systematic presentation of existing results in classical group theory to rhotrix group theory. Lastly, chapter five presents summary, conclusion and recommendations.
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