ON COSETS IN SPLIT EXTENSIONS OF HYPERCOMPLEX NUMBERS BY MUSYOKA DAVID MWANZIA REG NO. I56/CE/28760/2015 A Project submitted in partial fulfilment of the requirements for the Award of the Degree of Master of Science in Pure Mathematics in the School of Pure and Applied Sciences of Kenyatta University. June, 2019 i DECLARATION This project is my original work and has not been submitted for any degree award in any other university. Signature ……………………………………….. Date ………………………………………… MUSYOKA DAVID MWANZIA B. ED (Hons) This research project has been submitted for examination with my approval as a University Supervisor. Signature ……………………………………….. Date …………………………………………. DR. LYDIA N. NJUGUNA. MATHEMATICS DEPARTMENT KENYATTA UNIVERSITY. ii DEDICATION I would like to dedicate this work to my dear wife Fey and daughter Julie for their constant encouragement as I pursued my academic goals. I also dedicate my work to my parents and siblings for their unfailing support in my academic journey. iii ACKNOWLEDGEMENT First, I give glory to the Almighty God, I thank Him for the gift of life, good health and an opportunity to study. Indeed, all my efforts would have been in vain if He did not guide me. Secondly, I would like to acknowledge my supervisor Dr. Lydia N. Njuguna for her continued guidance especially in the field of Loop Theory, throughout the time I was preparing my project. She was always ready every week to guide me where necessary. I wish also to thank all my lecturers in the Department of Mathematics of Kenyatta University for having taught me in various units in Pure Mathematics. Lastly, I would like also to acknowledge my Pure Mathematics classmates Hannah, Dennis, Scholastica, Dalmas, Doris and Alex for their encouragement and useful criticism that helped better my work. iv TABLE OF CONTENTS DECLARATION ...................................................................................................................... ii DEDICATION ........................................................................................................................ iii ACKNOWLEDGEMENT ...................................................................................................... iv TABLE OF CONTENTS ..........................................................................................................v LIST OF TABLES ................................................................................................................. vii LIST OF SYMBOLS ............................................................................................................. viii ABSTRACT............................................................................................................................. ix CHAPTER ONE: INTRODUCTION ......................................................................................1 1.0 Introduction .......................................................................................................................1 1.1 Background Information ....................................................................................................1 1.1.1 The Cayley-Dickson doubling Process ........................................................................2 1.1.2 Multiplication of the split extension elements ..............................................................3 1.2 Statement of the problem ...................................................................................................3 1.3 Hypotheses ........................................................................................................................4 1.4 Objectives ..........................................................................................................................4 1.4.1 General objective.........................................................................................................4 1.4.2 Specific objectives .......................................................................................................4 1.5 Scope of the Study .............................................................................................................5 1.6 Significance of the study ....................................................................................................5 1.7 Definitions and Notations ..................................................................................................5 1.7.5 Left coset ........................................................................................................................6 1.7.6 Right coset ......................................................................................................................6 1.8 Nim Addition .....................................................................................................................8 1.8.1 Rules for Nim-addition ................................................................................................8 1.8.2 Properties of Nim addition:..........................................................................................9 CHAPTER TWO: LITERATURE REVIEW ........................................................................ 11 2. 0 Introduction .................................................................................................................... 11 2. 1 Hypercomplex numbers and their split extensions ........................................................... 11 2. 2 Properties of Cosets in Loops .......................................................................................... 12 2. 3 On Cosets in Steiner Loops ............................................................................................. 12 CHAPTER THREE: CYCLIC SUBLOOPS AND COSET DECOMPOSITION................ 14 3. 0 Introduction .................................................................................................................... 14 v 3. 1 Complex split extensions ................................................................................................ 14 3.1.1 Cyclic subloops of complex split extensions .............................................................. 16 3.1.2 Coset decomposition ................................................................................................. 17 3. 2 Quaternion split extensions ............................................................................................. 19 3.2.1 Cyclic subloops of quaternion split extensions ........................................................... 21 3.2.2 Coset decomposition ................................................................................................. 22 3. 3 Octonion split extensions ................................................................................................ 25 3.3.1 Cyclic subloops of octonion split extensions .............................................................. 27 3.3.2 Coset decomposition ................................................................................................. 29 3. 4 Sedenion split extension .................................................................................................. 32 3.4.1 Cyclic subloops of sedenion split extensions ............................................................. 33 3.4.2 Coset decomposition ................................................................................................. 35 CHAPTER FOUR: QUOTIENT LOOPS .............................................................................. 38 4.0 Introduction ..................................................................................................................... 38 4.1 Quotient loops arising from complex split extensions....................................................... 38 4.2 Quotient loops arising from quaternion split extension ..................................................... 40 4.3 Quotient loops arising from octonion split extensions ...................................................... 42 4.4 Quotient loops arising from sedenion split extensions ...................................................... 45 4.5 Conclusions ..................................................................................................................... 47 CHAPTER FIVE: CONCLUSIONS AND RECOMMENDATION FOR FURTHER RESEARCH ............................................................................................................................ 48 5.0 Introduction ..................................................................................................................... 48 5.1 Conclusions ..................................................................................................................... 48 5.2 Recommendation for future research ................................................................................ 48 REFERENCES ........................................................................................................................ 49 vi LIST OF TABLES Table 1.1: Nim Addition of the Numbers 0 to 31 ....................................................................... 10 Table 3.1: Multiplication of complex split extensions ................................................................ 15 Table 3.2: Multiplication of quaternion split extensions ............................................................. 20 Table 3.3: Multiplication of octonion split extensions................................................................ 26 Table 4.1.1 Multiplication of the elements of ℂ ⋊ 푆°/푆 ............................................................