An Algebraic Study of Non-commutative General linear Rhotrix Group
Chapter One
Aim and Objectives of the Study
The aim of this dissertation is to present an algebraic study of the development of non- commutative general linear rhotrix group. In particular, the following are the research objectives
- 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.
- To identify and study the properties of General Linear Rhotrix Group as analogous to the well-known General Linear Group in the
- To dissect the General Rhotrix Group in order to uncover its
- To establish the embedment of a particular subgroup of General Linear Rhotrix Group in a particular subgroup of General Linear
- To construct some finite non-commutative groups of rhotrices and identify their
CHAPTER TWO
LITERATURE REVIEW
Introduction
This chapter undertakes a review of existing literature in the theory of rhotrices starting from inception, 2003 up to the time when this dissertation was written. It considers the work of various researchers in the development of the theory of rhotrices. It reviews journal articles in both commutative and non-commutative rhotrix theories. It also considers the general linear group (group of matrices) as it will be seen later in chapter three to be analogues to the non-commutative general rhotrix group.
Rhotrix Theory
Rhotrix theory was initiated by Ajibade (2003) who defined rhotrix as a rhomboidal form of representing array of numbers. The concept is an extension of ideas on matrix-tersion and matrix-noitret proposed by Atanassov and Shannon (1998). Ajibade (2003) presented the initial concept, analysis and algebra of rhotrices, where he defined an operation of multiplication of rhotrices of size three. This operation of multiplication is known as heart-based multiplication and it satisfies the commutative property of binary operators.
Sani (2004) proposed an alternative method for multiplication of rhotrices of size three and later generalized the idea to rhotrices of size n in Sani (2007). This alternative method for rhotrix multiplication is known as row-column-based method for rhotrix multiplication and it is known to be non-commutative but associative.
Therefore, in the literature of rhotrix theory, two methods for multiplication of rhotrices having the same size are currently available. We have the heart-based method for rhotrix multiplication given by Ajibade (2003). This was followed by the row-column-based method for rhotrix multiplication proposed in Sani (2004), in an attempt to answer the question posed in Ajibade (2003) in the concluding section of his article. However, each of the two methods provides enabling environment to explore the usefulness of rhotrices as tools for carrying out mathematical research.
This chapter presents a comprehensive literature review of related articles in rhotrix theory and also gives associated literature on matrix theory. To achieve this, a classification of all the articles in the literature of rhotrix theory into two classes in line with the review of rhotrix theory carried out by Mohammed and Balarabe (2014) will be adopted. In their work, one class of the articles in the literature of rhotrix theory was termed as commutative rhotrix theory,while the other class was termed as non- commutative rhotrix theory. The reason behind their classification was due to the fact that contributory author(s) / researcher(s) in a single article, either adopted Ajibade (2003) heart-based method for multiplication of rhotrices or Sani (2004, 2007) row- column-based method for multiplication of rhotrices. Therefore, articles in literature adopting the heart-based method for rhotrix multiplication belong to the class ofcommutative rhotrix theory, while those articles in the literature adopting Sani‟s row- column-based method for rhotrix multiplication belong to theclass of non-commutative rhotrix theory.
CHAPTER THREE
THE NON-COMMUTATIVE RHOTRIX GROUPS
Introduction
This chapter considers the pair (GRn (F),∘) consisting of the set of all invertible rhotrices of size n over a field F; together with the binary operation of row-column-based method for rhotrix multiplication; ‘ ∘ ‘, in order tointroduce it as the concept of “non-commutative general linear rhotrix group”. We identify the subgroups of the (GRn (F),∘) andshow that any of its particular subgroups is embedded in a particular subgroup of the general linear group. Furthermore, an investigation of some isomorphic relationships between the subgroups in (GRn (F),∘) is made.
Non-commutative General Linear Rhotrix Group
In Sani (2007), it was stated as a remark (without proof) that the set of all invertible rhotrices of the same size with entries from the set of real numbers is a group with respect to row-column (non-commutative) method for rhotrix multiplication. In the following theorem, a generalization of non-commutative groups of rhotrices having the same size n with entries from an arbitrary field F is proposed.
CHAPTER FOUR
CONSTRUCTION OF SOME FINITE NON-COMMUTATIVERHOTRIX GROUPS
Introduction
In this chapter, we introduce concrete constructions of finite non-commutative rhotrix groups having entries from set of integers modulo p,where p is a positive prime.
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATIONS
Summary
We have considered an algebraic study of non-commutative rhotrix groups using rhotrix sets as underlying sets. In the process, a review of the progress made so far in the literature of rhotrix theory, starting from Ajibade (2003), when the concept was introduced, up to 2014 was made. A construction of non-commutative general linear rhotrix group considered to be analogous to the General Linear Group was presented.
The non-commutative general linear rhotrix group consists of all invertible rhotrices of size with entries from an arbitrary field F and it possesses all non-commutative rhotrix groups as its subgroups. Certain subgroups of non-commutative general linear rhotrix group were identified and then shown to be embedded in certain subgroups of the general linear group. Furthermore, some finite non-commutative groups of rhotrices as well as their subgroups were constructed and schematized.
Conclusion
In conclusion, we have presented new algebraic systems termed as Non-commutative General Linear Rhotrix Groups. Some finite and infinite non commutative rhotrix groups and their generalization were considered. A number of theorems had also been developed. It is our hope that this study will go to a large extent in simplification of teaching and learning of group theory in Mathematical discipline.
Recommendations
We recommend that the theory of rhotrix groups being a relatively new paradigm of Algebra be applied in a number of areas as follows:
- Computing non-commutative finite groups of rhotrices of larger size.
- Development of non-commutative finite cyclic groups of
- Construction and development of composition series for non-commutative finitegroup of rhotrices.
- Extension of Sylow theorems to non-commutative finite groups of
- Construction and development of non-commutative Polynomial groups of
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