On Cosets in Split Extensions of Hypercomplex Numbers

Home , Sedenion

ON COSETS IN SPLIT EXTENSIONS OF HYPERCOMPLEX

NUMBERS

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

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.

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.

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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.

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

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

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 ...... 38 Table 4.1.2 Multiplication of the elements of ℂ ⋊ 푆°/푀 ...... 39 Table 4.2.1 Multiplication of the elements of ℍ ⋊ 푆°/푆 ...... 40 Table 4.2.2 Multiplication of the elements of ℍ ⋊ 푆°/푀 ...... 41 Table 4.3.1 Multiplication of the elements of 𝕆 ⋊ 푆°/푆...... 43 Table 4.3.2 Multiplication of the elements of 𝕆 ⋊ 푆°/푀 ...... 44

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LIST OF SYMBOLS ⨁ -Nim addition

ℂ -Set of Complex Numbers

ℍ -Quaternion Algebra 𝕆 -Octonion Algebra

퐿 ⋊ 푆0-Sedenion extension of the loop L

푆0 -The set {1, −1}

𝕊 -Sedenion Algebra

∼ -Equivalence relation

⋊ -Right normal factor semidirect product

∕ -Divide

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ABSTRACT This work focuses on the study of some properties of cosets in split extensions of hypercomplex numbers. It is well known that if G is a group and H its subgroup, the cosets of the subgroup H form a partition of the group G. However, this property does not generally hold for loops. This study aims at constructing cyclic subloops of the split extensions of hypercomplex numbers and the corresponding cosets arising from them. It is then shown that the cosets of a cyclic subloop form a partition of the split extension loop i.e. any two right or left cosets of a cyclic subloop are either disjoint or identical. The study uses the Cayley-Dickson and Jonathan Smith doubling processes to construct multiplication tables for the split extensions of hypercomplex numbers. Nim addition is also used to give a general way of generating cyclic subloops and the cosets arising from them. In Loop Theory, only when 푆 is a normal subloop of 퐿 will the left and right cosets of 푆 coincide, these cosets form a loop 퐿/푆 called the quotient or factor loop whose multiplication is defined by (푎 ∙ 푆) ∙ (푏 ∙ 푆) = (푎 ∙ 푏) ∙ 푆, ∀ 푎, 푏 ∈ 퐿. In this work we use cyclic normal subloops of split extensions of hypercomplex numbers to construct quotient loops. Lastly, we show that the multiplication of the elements in the quotient loop formed can also be carried out by considering the Nim addition of the subscripts of the individual elements.

CHAPTER ONE: INTRODUCTION

1.0 Introduction

In this chapter we give the background information of the study, doubling formulas to be used, statement of the problem, hypotheses, general and specific objectives, scope of the study and definitions of some of the key words used in the study. We also define and give rules for Nim addition of non-negative integers.

1.1 Background Information

In Mathematics, a coset is a set made of all the products obtained from multiplication of every element of a subgroup H in turn by one element of the group G that contain the subgroup H.

Multiplication of an element of a group by the subgroup from the left gives rise to a left coset while multiplication from the right gives rise to a right coset. A coset may not necessarily be a subgroup of the group.

Cosets form the basis of this study. They play a very crucial role in proofs of some of the most basic results in Group Theory, for instance, in the proof of the Lagrange’s Theorem (Kinyon M,

Pula K and Vojtechovsky P, 2012). They are also used in construction of quotient loops. Recently,

Mathematicians have focused their attention on the study of cosets in Loop Theory. Michael

Kinyon, Kyle Pula and Petr Vojtechovsky studied the properties of cosets in Antiautomorphic loops and Bol loops (Kinyon M, Pula K and Vojtechovsky P, 2012). They showed that any two left cosets of a subloop 푆 of a left automorphic Moufang loop were either disjoint or intersect in a set whose number of elements equals that of some subloop of S. Ales Drapal and Terry Griggs gave a complete answer to the question of when the cosets of a Steiner subloop 푆 partition the loop

(Drapal A., Griggs T. S, 2016). They concluded that this happens if and only if the Steiner loop can be formed by a union of subloops of order 2|푆|, any two of which intersect in 푆.

1.1.1 The Cayley-Dickson doubling Process

The Cayley- Dickson doubling process starting from real numbers successively yields the complex numbers of dimension 2, quaternions of dimension 4, octonions of dimension 8, and sedenions of dimension 16 (Smith, W. D, 2004). Each algebra contains the previous one as a sub algebra.

Different doubling processes for obtaining sedenions from octonions and general 2푛-ons from

2푛−1-ons have been developed.

A complex number can be written in the form (푎, 푏) where 푎, 푏 ∈ ℝ. Doubling complex numbers by using the Cayley-Dickson process gives rise to a 22 – dimensional quaternion algebra. The multiplication is defined as follows:

(푎, 푏)(푐, 푑) = (푎푐 − 푏푑̅, 푏푐̅ + 푎푑) (1)

The 22 – dimensional quaternion algebra ℍ has a basis 1, 푖, 푗 and 푘. Therefore, if q ∈ ℍ, it can be written as, q = a + b푖 + c푗 + d푘, where a, b, c, d ∈ ℝ.

The multiplication for quaternions can be expressed as a rule i. e 푖2 = 푗2 = 푘2 = 푖푗푘 = −1 which implies that, 푖푗 = 푘, 푗푘 = 푖, 푘푖 = 푗, 푗푖 = −푘, 푘푗 = −푖, 푖푘 = −푗.

Next we have the 23-dimensional octonions algebra obtained by forming pairs (a, b) where 푎, 푏 ∈

ℍ and multiplication carried out as in complex numbers. In general, this process continues giving rise to 2푛-dimensional hypercomplex numbers from a pair of 2푛−1-dimensional hypercomplex numbers.

1.1.2 Multiplication of the split extension elements

The elements of 퐿 ⋊ 푆0 are encoded as pairs (푎, 푏) with 푎 ∈ 퐿 and 푏 ∈ 푆0 = {1, −1}. The multiplication of these elements will be done using the Jonathan Smith formula (Smith, J. D. H,

1995) given below:

i. (푥, 1)(푦, 1) = (푥푦, 1)

ii. (푥, 1)(푦, −1) = (푦푥, −1)

iii. (푥, −1)(푦, 1) = (푥푦̅, −1)

iv. (푥, −1)(푦, −1) = (−푥푦̅, 1) (2)

1.2 Statement of the problem

In Group Theory, it is expected that if 퐺 is a group and 퐻 its subgroup, then the left or right cosets of the subgroup 퐻 form a partition of the group i.e. the left or right cosets of a subgroup 퐻 partition the group into equivalence classes under the relation 푎~푏 where 푎, 푏 ∈ 퐺. However, the left or right cosets of a subloop may not partition a loop as is the case in a group. This is because, the proof that the left cosets of a subgroup partition the group uses all the properties of a group which may not all hold in a loop. For instance, the existence of identity element is used to prove reflexivity, the existence of inverses, associativity and identity element are used in the proof of transitivity. The aim of this study is to investigate whether the cosets of a cyclic subloop S of a split extension loop Q form a partition of the loop. The study also aims at constructing quotient loops using the cyclic normal subloops of the split extensions of hypercomplex numbers. Finally, it is also shown that the multiplication of the elements of the quotient loop formed can be viewed in terms of Nim addition of non-negative integers.

1.3 Hypotheses

In this study, we seek to answer the following hypotheses;

i. The cosets of a cyclic subloop S of a split extension loop Q forms a partition of the

loop Q.

ii. The cyclic subloop S of a split extension loop is normal.

iii. Quotient loops can be constructed using cyclic subloops of split extensions of hyper

complex numbers.

iv. Multiplication of the elements of the quotient loops can be carried out by considering

the Nim-addition of the subscripts of the individual elements.

1.4 Objectives

1.4.1 General objective

The study mainly aims at investigating some properties of cosets in the split extensions of hypercomplex numbers.

1.4.2 Specific objectives

The specific objectives of the study are to:

i. Construct cyclic subloops of split extensions of hypercomplex numbers.

ii. Show that the cosets of a cyclic subloop S of a split extension loop Q form a partition of

the loop Q. iii. Give a general way of generating cyclic subloops and cosets arising from them. iv. Construct quotient loops arising from the cosets of cyclic normal subloops and show that

the multiplication of the elements of the quotient loops can be performed by considering

the Nim-addition of the subscripts of the individual elements.

1.5 Scope of the Study

This study will focus on cosets in split extensions of hypercomplex numbers. The Cayley-Dickson and the Jonathan Smith doubling processes will be used to construct the multiplication tables for the split extensions of hypercomplex numbers. Cyclic subloops will be constructed by applying the loop operation repeatedly on a single element of the loop. Finally, Nim addition will be used to generalize the construction of the cyclic subloops, coset decomposition and multiplication of the elements of the quotient loops.

1.6 Significance of the study

The study of cosets in Group Theory plays a very important role especially in proof of Lagrange’s theorem and in construction of quotient groups. The study of the same in Loop Theory will help researchers to extend the proof of Lagrange’s theorem to different types of loops. It will also enable the construct factor loops.

1.7 Definitions and Notations

In this section, we give definitions and notations of some of the key words used throughout this study.

1.7.1 Groupoid A groupoid (푄, ∙) is a non-empty set 푄 which is closed under the binary operation ( ∙ ) i.e.

∀푥, 푦 ∈ 푄 ∃ 푧 ∈ 푄 such that 푥 ∙ 푦 = 푧.

1.7.2 Quasigroup A quasigroup is a groupoid (푄, ∙) such that ∀ 푎, 푏 ∈ 푄 the equations 푎푥 = 푏, and 푦푎 = 푏 have unique solutions 푥, 푦 ∈ 푄 respectively.

1.7.3 Loop A Loop is a quasigroup (푄, ∙) with a neutral element 푒 ∈ 푄 such that 푒푥 = 푥푒 = 푥 ∀푥 ∈ 푄. A loop with the associative property forms a group i.e. in Loop Theory a group is simply an associative loop.

1.7.4 Subloop A subloop is a non-empty subset 푆 ⊆ (푄, ∙) denoted by 푆 ≤ 푄 such that (푆, ∙) is a loop in its own right.

1.7.5 Left coset For a loop 푄, a subloop 푆 ≤ 푄, and 푥 ∈ 푄, then the subset 푥푆 = {푥푠: 푠 ∈ 푆} ⊆ 푄 is the left coset of 푆 containing 푥.

1.7.6 Right coset For a loop 푄, a subloop 푆 ≤ 푄, and 푥 ∈ 푄, then the subset 푆푥 = {푠푥: 푠 ∈ 푆} ⊆ 푄 is the right coset of 푆 containing 푥.

1.7.7 Commutant The commutant, 퐶(푄), of a loop 푄 is the set of those elements 푐 ∈ 푄 which commute with each element in the loop. That is, 퐶(푄) = {푐 ∈ 푄 ∶ ∀푥 ∈ 푄, 푐푥 = 푥푐}

1.7.8 Decomposition property A loop 푄 has left (respectively right) coset decomposition modulo S if the set of all left

(respectively right) cosets modulo S is a partition of Q (Drapal A., Griggs T. S, 2016).

1.7.9 Bol loop A Bol loop is loop (푄, ∙) satisfying the left or right Bol laws i.e. ∀ 푥, 푦, 푧 ∈ 푄:

(i) (푥 ∙ 푦푥)푧 = 푥(푦 ∙ 푥푧) Left Bol law,

(ii) 푧(푥푦 ∙ 푥) = (푧푥 ∙ 푦)푥) Right Bol law.

1.7.10 Moufang loop A loop L is called a Moufang loop if it satisfies any of the following equivalent identities:

(i) 푥푦. 푧푥 = (푥. 푦푧)푥,

(ii) 푥(푦. 푥푧) = (푥푦. 푥)푧,

(iii)푥(푦. 푧푦) = (푥푦. 푧)푦 for all 푥, 푦, 푧 ∈ 퐿.

1.7.11 Steiner loop A Steiner loop or a sloop is a groupoid (퐿,∙, 푒), where ( ∙ ) is a binary operation and 푒 is a constant satisfying the identities:

(i) 푒 ∙ 푥 = 푥,

(ii) 푥 ∙ 푦 = 푦 ∙ 푥,

(iii)푥 ∙ (푥 ∙ 푦) = 푦 for all 푥, 푦 ∈ 퐿.

1.7.12 Antiautomorphic loop A loop L is said to be an antiautomorphic loop if it satisfies the following property

푥푦̅̅̅ = 푦̅푥̅ ∀ 푥, 푦 ∈ 퐿.

1.7.13 Normal Subloop

A subloop 푆 of a loop 퐿 is said to be normal if for all 푎, 푏 ∈ 퐿, the following holds:

(푎 ∙ 푏) ∙ 푆 = 푎 ∙ (푏 ∙ 푆) = 푎 ∙ (푆 ∙ 푏)

Note that the equality of the first two is not guaranteed because we do not assume the loop to be associative.

1.7.14 Quotient loop Let (퐿,∙) be a loop and 푆 a normal subloop of 퐿. The quotient loop 퐿/푆 is defined as the following loop:

1. The set of elements of 퐿/푆 is the set of left cosets of 푆,i.e. subsets of the form

푎 ∙ 푆, with 푎 ∈ 퐿.

2. The multiplication is defined by (푎 ∙ 푆) ∙ (푏 ∙ 푆) = (푎 ∙ 푏) ∙ 푆 and is well defined

and follows from the definition of a normal subloop.

1.8 Nim Addition

Nim addition gives a convenient way of defining addition in ℤ+ to make it a field of characteristic two. It is carried out by first writing the individual numbers in binary form and then adding without carrying over.

1.8.1 Rules for Nim-addition

i. The Nim sum of a number of distinct 2-powers is their ordinary sum.

Example

64 ⨁ 32 ⨁ 16 = 1000000 ⨁ 100000 ⨁ 10000 = 1110000 = 104 = 64 + 32 + 16

ii. The Nim sum of two equal numbers is zero.

Example

64 ⨁ 64 = 1000000 ⨁ 1000000 = 0000000 = 0

In this study we use Nim addition to:

i. Generalize the construction of a cyclic subloop of the split extensions of hypercomplex

numbers.

ii. Give a general way of generating left and right cosets of a subloop. iii. Show that the multiplication of the elements of a quotient loop is similar to Nim addition

of the subscripts of the individual elements.

The following table gives the Nim-addition of the elements 0, 1, 2, …,31.

1.8.2 Properties of Nim addition:

(a) 훼 ⨁ 훽 = 훽 ⨁ 훼

(b) 훼 ⨁ 0 = 0 ⨁ 훼 = 훼

(d) (훼 ⨁ 훽) ⊕ 훾 = 훼 ⨁(훽 ⊕ 훾)

Table 1.1: Nim Addition of the Numbers 0 to 31 (Kivunge B. M, 2004)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 2 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 3 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 19 18 17 16 23 22 21 20 27 26 25 24 31 30 29 28 4 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 20 21 22 23 16 17 18 19 28 29 30 31 24 25 26 27 5 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 21 20 23 22 17 16 19 18 29 28 31 30 25 24 27 26 6 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 22 23 20 21 18 19 16 17 30 31 28 29 26 27 24 25 7 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 23 22 21 20 19 18 17 16 31 30 29 28 27 28 25 24 8 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 24 25 26 27 28 29 30 31 16 17 18 19 20 21 22 23 9 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 25 24 27 26 29 28 31 30 17 16 19 18 21 20 23 22 10 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 26 27 24 25 30 31 28 29 18 19 16 17 22 23 20 21 11 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 27 26 25 24 31 30 29 28 19 18 17 16 23 22 21 20 12 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 28 29 30 31 24 25 26 27 20 21 22 23 16 17 18 19 13 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 29 28 31 30 25 24 27 26 21 20 23 22 17 16 19 18 14 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 30 31 28 29 26 27 24 25 22 23 20 21 18 19 16 17 15 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 18 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 19 19 18 17 16 23 22 21 20 27 26 25 24 31 30 29 28 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 20 20 21 22 23 16 17 18 19 28 29 30 31 24 25 26 27 4 5 6 7 0 1 2 3 4 13 14 15 8 9 10 11 21 21 20 23 22 17 16 19 18 29 28 31 30 25 24 27 26 5 4 7 6 1 0 3 2 13 12 14 15 9 8 11 10 22 22 23 20 21 18 19 16 17 30 31 28 29 26 27 24 25 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 23 23 22 21 20 19 18 17 16 31 30 29 28 27 26 25 24 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 24 24 25 26 27 28 29 30 31 16 17 18 19 20 21 22 23 8 9 10 11 4 13 14 15 0 1 2 3 4 5 6 7 25 25 24 27 26 29 28 31 30 17 16 19 18 21 20 23 22 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 26 26 27 24 25 30 31 28 29 18 19 16 17 22 23 20 21 10 11 8 9 14 14 12 13 2 3 0 1 6 7 4 5 27 27 26 25 24 31 20 29 28 19 18 17 16 23 22 21 20 11 10 9 8 15 15 13 12 3 2 1 0 7 6 5 4 28 28 29 30 31 24 25 28 27 20 21 22 23 16 17 18 19 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 29 29 28 31 30 25 24 27 28 21 20 23 22 17 16 19 18 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 30 30 31 28 29 26 27 24 25 22 23 20 21 18 19 16 17 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 31 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

CHAPTER TWO: LITERATURE REVIEW

2. 0 Introduction

In this chapter, we give a literature review of previous research on the study of hypercomplex numbers and their split extensions. We also review previous research on the study of cosets in finite loops.

2. 1 Hypercomplex numbers and their split extensions

Kivunge (2004) studied hypercomplex numbers with major emphasis on the 16 dimensional sedenions. The researcher introduced sedenion extensions and gave equivalent conditions for the extensions to be groups. It was also showed that the frame multiplication of the 2푛- ons can be viewed in terms of designs, Nim addition and projective geometry.

Magero (2007) constructed loops abstractly as extensions of subloops of quaternions, octonions and sedenions. The researcher also studied the properties that hold for the split extensions of hypercomplex numbers and gave respective specific examples. Among the properties studied were, Jordan identity, flexible property, right alternative property, left alternative property and antiautomorphic property.

Macharia (2008) studied the loop properties that hold for basal elements of 2푛 −dimensional hypercomplex numbers for 푛 = 1, 2, 3 푎푛푑 4 with major emphasis on the 16 −dimensional sedenions.

Kimeu (2008) studied some properties of hypercomplex number basal elements. The author constructed sign matrices of products and Fano planes and related the products to Nim addition.

Nim addition was also used to multiply the basal elements of hypercomplex numbers.

Njuguna (2012) worked on the sedenion extension loops and frames of hypercomplex numbers.

The researcher investigated the properties of the split extensions 퐿 ⋊ 푆0 particularly with 퐿 being non abelian. A theorem that showed how the multiplication of split extensions for 2푛 − 표푛푠 can be viewed in terms of Nim addition was also developed.

Masila (2015) constructed walsh functions from hadamard matrices arising from frame multiplication tables of hypercomplex numbers. The author remarked that Walsh functions are the only known functions with desirable features comparable to sine and cosine functions for use in communication.

2. 2 Properties of Cosets in Loops

Kinyon, Pula and Vojtechovsky (2012) studied properties of cosets in finite loops. They mainly focused their study on antiautomorphic loops and Bol loops. Their motivation was twofold. First they wanted to find an elementary proof of the Lagrange’s theorem using cosets and secondly, they were motivated by the possibility of realizing interesting combinatorial designs of cosets in loops.

2. 3 On Cosets in Steiner Loops

Ales Drapal and Terry Griggs (2016) studied cosets in Steiner loops. They were interested in giving a complete answer to the question of when the cosets of a Steiner subloop partition the loop. They concluded that the cosets of a Steiner subloop 푆 partition the loop if and only if the Steiner loop can be formed by a union of subloops of order 2|푆|, any two of which intersect in 푆.

From the review of literature, it is evident that Mathematicians have focused their attention on the study of hypercomplex numbers, their split extensions and on the properties of cosets in different types of loops. However, there exists no study on the cosets in the split extensions of hypercomplex numbers. This study seeks to investigate some properties of cosets in the split extensions of hypercomplex numbers.

CHAPTER THREE: CYCLIC SUBLOOPS AND COSET DECOMPOSITION

3. 0 Introduction

In this Chapter, we use the Cayley-Dickson and the Jonathan Smith doubling processes to construct multiplication tables of the split extensions of hypercomplex numbers. Cyclic subloops are constructed by repeatedly applying the loop operation on a single element and general way of generating the cyclic subloops by Nim addition is given. We also perform coset decomposition and determine whether the left or right cosets of a cyclic subloop partition the split extension loop.

Finally, Nim addition is used to generalize coset decomposition.

3. 1 Complex split extensions

Real numbers are a 2-dimensional algebra and they consist of rational numbers and irrational numbers. A complex number 푧 has the form 푎 + 푏푖 where 푎, 푏 ∈ ℝ and 푖 is the imaginary unit with the property 푖2 = −1. Complex numbers form the basis for other hypercomplex numbers.

The basis of a complex number consists of 1 and 푖.

For complex split extensions, the basal elements are defined as follows:

훼0 = {(1,1), 훼1 = (푖, 1), 훼2 = (−1,1), 훼3 = (−푖, 1), 훼4 = (−1, −1), 훼5 = (−푖, −1),

훼6 = (1, −1), 훼7 = (푖, −1)}

The multiplication of these elements is given by the following table.

Table 3.1: Multiplication of complex split extensions (Magero F.B, 2007)

훼0 훼1 훼2 훼3 훼4 훼5 훼6 훼7

훼0 훼0 훼1 훼2 훼3 훼4 훼5 훼6 훼7

훼1 훼1 훼2 훼3 훼0 훼5 훼6 훼7 훼4

훼2 훼2 훼3 훼0 훼1 훼6 훼7 훼4 훼5

훼3 훼3 훼0 훼1 훼2 훼7 훼4 훼5 훼6

훼4 훼4 훼7 훼6 훼5 훼2 훼1 훼0 훼3

훼5 훼5 훼4 훼7 훼6 훼3 훼2 훼1 훼0

훼6 훼6 훼5 훼4 훼7 훼0 훼3 훼2 훼1

훼7 훼7 훼6 훼5 훼4 훼1 훼0 훼3 훼2

Observations

1. Each element appears only once in every row and every column. Thus any two elements

define a third one uniquely i.e. ℂ ⋊ 푆° is a quasigroup.

2. 훼0 is the two-sided identity for ℂ ⋊ 푆° i.e 훼0훼푖 = 훼푖훼0 = 훼푖 for all 푖 = {0,1,2, … ,7}

3. ℂ ⋊ 푆° is associative since

(훼푖 훼푗)훼푘 = 훼푖 (훼푗훼푘) ∀ 푖, 푗, 푘 = {0,1,2, … ,7}

Example 3.1.1

(훼1 훼2)훼3 = 훼3. 훼3 = 훼2 = 훼1. 훼1 = 훼1 (훼2훼3)

(훼2 훼4)훼7 = 훼6. 훼7 = 훼1 = 훼2. 훼3 = 훼2 (훼4훼7)

3.1.1 Cyclic subloops of complex split extensions The cyclic subloop generated by 훼1 ∈ ℂ ⋊ 푆° is denoted by 〈훼1〉 and its elements are generated as follows:

1 1 훼1 = (푖, 1) = (푖, 1) = 훼1

2 2 훼1 = (푖, 1) = (푖, 1)(푖, 1) = (푖. 푖, 1) = (−1, 1) = 훼2 3 3 훼1 = (푖, 1) = (푖, 1)(−1, 1) = (푖. −1, 1) = (−푖, 1) = 훼3 4 4 훼1 = (푖, 1) = (−푖, 1)(푖, 1) = (−푖. 푖, 1) = (1, 1) = 훼0 Thus,

〈훼1〉 = {훼1 , 훼2 , 훼3 , 훼0} = 〈훼3〉

Similarly, we can have other cyclic subloops generated by other elements:

1. 〈훼2〉 = {훼2 , 훼0}

2. 〈훼4〉 = {훼4 , 훼2 , 훼6 , 훼0} = 〈훼6〉

3. 〈훼5〉 = {훼5 , 훼2 , 훼7 , 훼0} = 〈훼7〉

Remark 3.1.1.1 The commutant of ℂ ⋊ 푆° is 〈훼2〉 = {훼2 , 훼0} with, |〈훼2〉| = 2.

Theorem 3.1.1 Let 〈훼푖〉 be a cyclic subloop of ℂ ⋊ 푆° then:

1. The elements of the cyclic subloop 〈훼푖〉 can be generated using Nim addition as follows:

〈훼푖〉 = {훼푖, 훼푐, 훼푐⊕ 푖, 훼0}

where, 훼푖 is the generating element, for all 푖 ∈ {0,1, … ,7}, and 훼푐 is the non-identity element in the commutant, 퐶(ℂ ⋊ 푆° ).

2. 〈훼푖〉 = 〈훼푖 ⨁ 푐〉 ∀ 훼푖 ∈ ℂ ⋊ 푆° and 푐 the subscript for the non-identity element in

퐶(ℂ ⋊ 푆° ) provided that 푖 ≠ 0.

3.1.2 Coset decomposition We now construct distinct left and right cosets using the cyclic subloops in section 3.1.1 above.

Examples 3.1.2.1

1. If 푆 = 〈훼2〉 = {훼2 , 훼0}, then the left and the right cosets are:

훼1푆 = 훼1{훼2 , 훼0} = {훼3, 훼1 } = 푆훼1

훼4푆 = 훼4{훼2 , 훼0} = {훼6, 훼4 } = 푆훼4

훼5푆 = 훼5{훼2 , 훼0} = {훼7, 훼5} = 푆훼5

2. If 푆 = 〈훼1〉 = {훼1 , 훼2 , 훼3 , 훼0} = 〈훼3〉, then,

훼4푆 = 훼4{훼1 , 훼2 , 훼3 , 훼0} = {훼7, 훼6 , 훼5 , 훼4}

푆훼4 = {훼1 , 훼2 , 훼3 , 훼0}훼4 = {훼5, 훼6 , 훼7 , 훼4}

3. If 푆 = 〈훼4〉 = {훼4 , 훼2 , 훼6 , 훼0} = 〈훼6〉, then,

훼1푆 = 훼1{훼4 , 훼2 , 훼6 , 훼0} = {훼5, 훼3 , 훼7 , 훼1}

푆훼1 = {훼4 , 훼2 , 훼6 , 훼0}훼1 = {훼7, 훼3 , 훼5 , 훼1}

4. If 푆 = 〈훼5〉 = {훼5 , 훼2 , 훼7 , 훼0} = 〈훼7〉 then,

훼1푆 = 훼1{훼5 , 훼2 , 훼7 , 훼0} = {훼6 , 훼3 , 훼4 , 훼1}

푆훼1 = {훼5 , 훼2 , 훼7 , 훼0}훼1 = {훼4, 훼3 , 훼6 , 훼1}

Theorem 3.1.2 Let 푆 = 〈훼푖〉 = {훼푖, 훼푐, 훼푐 ⊕ 푖, 훼0} be a cyclic subloop, to get the left cosets of 푆, we choose 훼푗 ∈ ℂ ⋊ 푆° such that 훼푗 ∉ 푆 and proceed as follows:

훼푗푆 = {훼푗⊕푖, 훼푗⊕푐, 훼j⊕(푐 ⊕푖), 훼푗⊕0}

Similarly, to get the right cosets of S,

푆훼푗 = {훼푖⨁푗, 훼푐⨁푗 , 훼(푐⨁푖)⨁푗 , 훼0⨁푗}

Example 3.1.2.2

Let 푆 = 〈훼5〉 = {훼5 , 훼2 , 훼7 , 훼0} = 〈훼7〉 then the subscripts for the elements of 훼1푆 are given by:

훼1푆 = 훼1{훼5 , 훼2 , 훼7 , 훼0}

= {훼1훼5, 훼1훼2, 훼1훼7, 훼1훼0}

= {훼1 ⨁ 5, 훼1 ⨁ 2, 훼1 ⨁ 7, 훼1⨁0}

= {훼4, 훼3, 훼6, 훼1}

Conclusions

Given ∀ 훼푖 , 훼푗 ∈ ℂ × 푆°, 푖, 푗 ∈ {0, … ,7}

(i) 훼푖 ∈ 훼푖푆 or 훼푖 ∈ 푆훼푖

(ii) 훼푖푆 = 훼푗푆 and 훼푖푆 ∩ 훼푗푆 = ∅ or 푆훼푖 = 푆훼푗 and 푆훼푖 ∩ 푆훼푗 = ∅

(iii) |훼푖푆| = |훼푗푆| or |푆훼푖| = |푆훼푗|

Therefore, the left or right cosets of a cyclic subloop S partition ℂ ⋊ 푆° into equivalence classes under the relation 훼푖~훼푗.

3. 2 Quaternion split extensions

Quaternions ℍ form a four dimensional algebra with the basis 1, 푖, 푗, and 푘. The multiplication of these basis elements is defined by:

푖2 = 푗2 = 푘2 = 푖푗푘 = −1, 푖푗 = 푘, 푗푘 = 푖, 푘푖 = 푗, 푗푖 = −푘, 푘푗 = −푖, and 푖푘 = −푗.

The quaternion split extensions ℍ ⋊ 푆° forms a set of order 16. The basal elements of ℍ ⋊ 푆° are defined as follows

훽0 = (1,1), 훽1 = (푖, 1), 훽2 = (푗, 1), 훽3 = (푘, 1), 훽4 = (−1, −1), 훽5 = (−푖, −1), 훽6 = (−푗, −1),

훽7 = (−푘, −1), 훽8 = (−1,1), 훽9 = (−푖, 1), 훽10 = (−푗, 1), 훽11 = (−푘, 1), 훽12 = (1, −1),

훽13 = (푖, −1), 훽14 = (푗, −1), 훽15 = (푘, −1)

The multiplication of these elements is given by the following table:

Table 3.2: Multiplication of quaternion split extensions (Magero F.B, 2007)

훽0 훽1 훽2 훽3 훽4 훽5 훽6 훽7 훽8 훽9 훽10 훽11 훽12 훽13 훽14 훽15

훽0 훽0 훽1 훽2 훽3 훽4 훽5 훽6 훽7 훽8 훽9 훽10 훽11 훽12 훽13 훽14 훽15

훽1 훽1 훽8 훽3 훽10 훽5 훽12 훽15 훽6 훽9 훽0 훽11 훽2 훽13 훽4 훽7 훽14

훽2 훽2 훽11 훽8 훽1 훽6 훽7 훽12 훽13 훽10 훽3 훽0 훽9 훽14 훽15 훽4 훽5

훽3 훽3 훽2 훽9 훽8 훽7 훽14 훽5 훽12 훽11 훽10 훽1 훽0 훽15 훽6 훽13 훽4

훽4 훽4 훽13 훽14 훽15 훽8 훽1 훽2 훽3 훽12 훽5 훽6 훽7 훽0 훽9 훽10 훽11

훽5 훽5 훽4 훽15 훽6 훽9 훽8 훽3 훽10 훽13 훽12 훽7 훽14 훽1 훽0 훽11 훽2

훽6 훽6 훽7 훽4 훽13 훽10 훽11 훽8 훽1 훽14 훽15 훽12 훽5 훽2 훽3 훽0 훽9

훽7 훽7 훽14 훽5 훽4 훽11 훽2 훽9 훽8 훽15 훽6 훽13 훽12 훽3 훽10 훽1 훽0

훽8 훽8 훽9 훽10 훽11 훽12 훽13 훽14 훽15 훽0 훽1 훽2 훽3 훽4 훽5 훽6 훽7

훽9 훽9 훽0 훽11 훽2 훽13 훽4 훽7 훽14 훽1 훽8 훽3 훽10 훽5 훽12 훽15 훽6

훽10 훽10 훽3 훽0 훽9 훽14 훽15 훽4 훽5 훽2 훽11 훽8 훽1 훽6 훽7 훽12 훽13

훽11 훽11 훽10 훽1 훽0 훽15 훽6 훽13 훽4 훽3 훽2 훽9 훽8 훽7 훽14 훽5 훽12

훽12 훽12 훽5 훽6 훽7 훽0 훽9 훽10 훽11 훽4 훽13 훽14 훽15 훽8 훽1 훽2 훽3

훽13 훽13 훽12 훽7 훽14 훽1 훽0 훽11 훽2 훽5 훽4 훽15 훽6 훽9 훽8 훽3 훽10

훽14 훽14 훽15 훽12 훽5 훽2 훽3 훽0 훽9 훽6 훽7 훽4 훽13 훽10 훽11 훽8 훽1

훽15 훽15 훽6 훽13 훽12 훽3 훽10 훽1 훽0 훽7 훽14 훽5 훽4 훽11 훽2 훽9 훽8

Observations

1. Each element appears only once in every row and every column. Thus any two elements

define a third one uniquely i.e. ℍ ⋊ 푆° is a quasigroup.

2. 훽0 is the two-sided identity i.e. 훽0훽푖 = 훽푖훽0 = 훽푖 for all 푖 = {0,1,2, … ,15}, and hence

ℍ ⋊ 푆° forms a loop.

3. ℍ ⋊ 푆° is not associative since

(훽푖훽푗)훽푘 ≠ 훽푖(훽푗훽푘) ∀푖, 푗, 푘 = {0,1,2, … ,15}

Example 3.2.1

(훽9훽11)훽12 = 훽10. 훽12 =훽6 ≠ 훽9(훽11훽12) = 훽9. 훽7 = 훽14

{(−푖, 1)(−푘, 1)}(1, −1) = (푖푘, 1)(1, −1) = (−푗, −1) ≠ (−푖, 1){(−푘, 1)(1, −1)}

= (−푖, 1)(−푘, −1) = (푘푖, −1) = (푗, −1)

3.2.1 Cyclic subloops of quaternion split extensions

In this section we construct cyclic subloops generated by each element in quaternion split extensions. For example, the cyclic subloop generated by 훽1, which is denoted by 〈훽1〉 can be constructed as follows.

1 1 〈훽1〉 = (푖, 1) = (푖, 1) = 훽1

2 〈훽1〉 = (푖, 1)(푖, 1) = (−1,1) = 훽8

3 〈훽1〉 = (−1,1)(푖, 1) = (−푖, 1) = 훽9

4 〈훽1〉 = (−푖, 1)(푖, 1) = (1,1) = 훽0

Thus,

〈훽1〉 = {훽1, 훽8, 훽9, 훽0} = 〈훽9〉

Similarly, we can construct other cyclic subloops generated by other elements,

1. 〈훽2〉 = {훽2, 훽8, 훽10, 훽0} = 〈훽10〉

2. 〈훽3〉 = {훽3, 훽8, 훽11, 훽0} = 〈훽11〉 21

3. 〈훽4〉 = {훽4, 훽8, 훽12, 훽0} = 〈훽12〉

4. 〈훽5〉 = {훽5, 훽8, 훽13, 훽0} = 〈훽13〉

5. 〈훽6〉 = {훽6, 훽8, 훽14, 훽0} = 〈훽14〉

6. 〈훽7〉 = {훽7, 훽8, 훽15, 훽0} = 〈훽15〉

7. 〈훽8〉 = {훽8, 훽0}

Remark 3.2.1.1 In this case, 〈훽8〉 = {훽8, 훽0} is the commutant of ℍ ⋊ 푆° with |〈훽8〉| = 2

Theorem 3.2.1 Let 〈훽푖〉 be a cyclic subloop of ℍ ⋊ 푆°, then:

1. The elements of the cyclic subloop can be generated using Nim addition as follows:

〈훽푖〉 = {훽푖, 훽푐, 훽푐⨁푖, 훽0}

where, 훽푖 is the generating element, for all 푖 ∈ {0,1, … ,15}, 훽푐 is the non-identity element in the commutant, 퐶(ℍ ⋊ 푆°)

0 2. 〈훽푖〉 = 〈훽푖 ⨁ 푐〉 ∀ 훽푖 ∈ ℍ ⋊ 푆 and 푐 the subscript for the non-identity element in

퐶(ℍ ⋊ 푆°) provided that 푖 ≠ 0

3.2.2 Coset decomposition

We now construct distinct left and right cosets of some cyclic subloops in section 3. 2. 1 above.

Examples 3.2.2.1

1. Let S = 〈훽8〉 = {훽8, 훽0} then the left cosets of 푆 are as follows:

훽1푆 = 훽1{훽8, 훽0} = {훽9, 훽1} = {훽8, 훽0}훽1 = 푆훽1

훽2푆 = 훽2{훽8, 훽0} = {훽10, 훽2} = {훽8, 훽0}훽2 = 푆훽2

훽3푆 = 훽3{훽8, 훽0} = {훽11, 훽3} = {훽8, 훽0}훽3 = 푆훽3

훽4푆 = 훽4{훽8, 훽0} = {훽12, 훽4} = {훽8, 훽0}훽4 = 푆훽4

훽5푆 = 훽5{훽8, 훽0} = {훽13, 훽5} = {훽8, 훽0}훽5 = 푆훽5

훽6푆 = 훽6{훽8, 훽0} = {훽14, 훽6} = {훽8, 훽0}훽6 = 푆훽6

훽7푆 = 훽7{훽8, 훽0} = {훽15, 훽7} = {훽8, 훽0}훽7 = 푆훽7

2. If 푆 = 〈훽1〉 = {훽1, 훽8, 훽9, 훽0} = 〈훽9〉 then

훽2푆 = 훽2{훽1, 훽8, 훽9, 훽0} = {훽11, 훽10, 훽3, 훽2} ,

훽4푆 = 훽4{훽1, 훽8, 훽9, 훽0} = {훽13, 훽12, 훽5, 훽4},

훽6푆 = 훽6{훽1, 훽8, 훽9, 훽0} = {훽7, 훽14, 훽15, 훽6}

푆훽2 = {훽1, 훽8, 훽9, 훽0}훽2 = {훽3, 훽10, 훽11, 훽2} ,

푆훽4 = {훽1, 훽8, 훽9, 훽0}훽4 = {훽5, 훽12, 훽13, 훽4},

푆훽6 = {훽1, 훽8, 훽9, 훽0}훽6 = {훽15, 훽14, 훽7, 훽6}

3. If 푆 = 〈훽2〉 = {훽2, 훽8, 훽10, 훽0} = 〈훽10〉 then,

훽1푆 = 훽1{훽2, 훽8, 훽10, 훽0} = {훽3, 훽9, 훽11, 훽1} ,

훽4푆 = 훽4{훽2, 훽8, 훽10, 훽0} = {훽14, 훽12, 훽6, 훽4},

훽5푆 = 훽5{훽2, 훽8, 훽10, 훽0} = {훽15, 훽13, 훽7, 훽5}

푆훽1 = {훽2, 훽8, 훽10, 훽0}훽1 = {훽11, 훽9, 훽3, 훽1} ,

푆훽4 = {훽2, 훽8, 훽10, 훽0}훽4 = {훽6, 훽12, 훽14, 훽4},

푆훽5 = {훽2, 훽8, 훽10, 훽0}훽5 = {훽7, 훽14, 훽15, 훽5}

Following the same method, the left and right cosets generated by the other cyclic subloops of the quaternion split extensions can be obtained.

Theorem 3.2.2 Let 푆 = 〈훽푖〉 = {훽푖, 훽푐, 훽푐⨁ 푖, 훽0} be a cyclic subloop of ℍ ⋊ 푆° ,to get the left cosets of 푆, we choose an element 훽푗 ∈ ℍ ⋊ 푆° such that 훽푗 ∉ 푆 and carry out Nim addition as follows:

훽푗푆 = {훽(푗⨁푖), 훽(푗⨁ 푐), 훽푗⨁(푐⨁푖), 훽(푗⨁0)}

On the other hand, to get the right cosets of 푆 = 〈훽푖〉 = {훽푖, 훽푐, 훽푐⨁ 푖, 훽0, we have,

푆훽푗 = {훽(푖⨁푗), 훽(푐⨁ 푗), 훽(푐⨁푖)⨁푗, 훽(0⨁푗)}

Example 3.2.2.2

Let 푆 = 〈훽7〉 = {훽7, 훽8, 훽15, 훽0} then the subscripts for the elements of 훽3푆 are obtained by Nim addition as follows:

훽3푆 = 훽3{훽7, 훽8, 훽15, 훽0}

= {훽3훽7, 훽3훽8, 훽3훽15, 훽3훽0}.

= {훽3⨁7, 훽3⨁8, 훽3⨁15, 훽3⨁0}

= {훽4, 훽11, 훽12, 훽3}

Conclusions

Given ∀ 훽푖, 훽푗 ∈ ℍ ⋊ 푆°, 푖, 푗 ∈ {0, … ,15}

(i) 훽푖 ∈ 훽푖푆 or 훽푖 ∈ 푆훽푖

(ii) 훽푖푆 = 훽푗 푆 and 훽푖푆 ∩ 훽푗푆 = ∅ or 푆훽푖 = 푆훽푗 and 푆훽푖 ∩ 푆훽푗 = ∅

(iii) |훽푖푆| = |훽푗푆| 표푟|푆훽푖| = |푆훽푗 |

Therefore, the left or right cosets of a cyclic subloop 푆 partition ℍ ⋊ 푆° into equivalence classes under the relation 훽푖~훽푗.

3. 3 Octonion split extensions

3 Octonions 𝕆 form an 2 -dimension algebra whose basis elements are, {푒0 = (1,0), 푒1 = (푖, 0),

푒2 = (푗, 0), 푒3 = (푘, 0), 푒4 = (0,1), 푒5 = (0, 푖), 푒6 = (0, 푗), 푒7 = (0, 푘)}. The multiplication of these elements will be carried out using the Cayley Dickson process.

The Octonion Split Extensions form a set of order 32. Its elements are of the form (푥, 푦) where

푥 ∈ 𝕆 and 푦 ∈ 푆°. The elements of 𝕆 ⋊ 푆° are thus defined as follows:

휇0 = (푒0, 1), 휇1 = (푒1, 1), 휇2 = (푒2, 1) , 휇3 = (푒3, 1) , 휇4 = (푒4, 1) , 휇5 = (푒5, 1),

휇6 = (푒6, 1), 휇7 = (푒7, 1), 휇8 = (−푒0, 1), 휇9 = (−푒1, 1), 휇10 = (−푒2, 1), 휇11 = (−푒3, 1),

휇12 = (−푒4, 1), 휇13 = (−푒5, 1), 휇14 = (−푒6, 1) , 휇15 = (−푒7, 1), 휇16 = (푒0, −1),

휇17 = (푒1, −1), 휇18 = (푒2, −1), 휇19 = (푒3, −1), 휇20 = (푒4, −1), 휇21 = (푒5, −1),

휇22 = (푒6, −1), 휇23 = (푒7, −1), 휇24 = (−푒0, −1), 휇25 = (−푒1, −1), 휇26 = (−푒2, −1),

휇27 = (−푒3, −1), 휇28 = (−푒4, −1), 휇29 = (−푒5, −1), s휇30 = (−푒6, −1), 휇31 = (−푒7, −1)

The multiplication table of these elements is given by the following table:

Table 3.3: Multiplication of octonion split extensions (Magero F.B, 2007)

휇0 휇1 휇2 휇3 휇4 휇5 휇6 휇7 휇8 휇9 휇10 휇11 휇12 휇13 휇14 휇15 휇16 휇17 휇18 휇19 휇20 휇21 휇22 휇23 휇24 휇25 휇26 휇27 휇28 휇29 휇30 휇31 휇0 휇0 휇1 휇2 휇3 휇4 휇5 휇6 휇7 휇8 휇9 휇10 휇11 휇12 휇13 휇14 휇15 휇16 휇17 휇18 휇19 휇20 휇21 휇22 휇23 휇24 휇25 휇26 휇27 휇28 휇29 휇30 휇31 휇1 휇1 휇8 휇3 휇10 휇5 휇12 휇15 휇6 휇9 휇0 휇11 휇2 휇13 휇4 휇7 휇14 휇17 휇24 휇27 휇18 휇29 휇20 휇23 휇30 휇25 휇16 휇19 휇26 휇21 휇28 휇31 휇22 휇2 휇2 휇11 휇8 휇1 휇6 휇7 휇12 휇13 휇10 휇3 휇0 휇9 휇14 휇15 휇4 휇5 휇18 휇19 휇24 휇25 휇30 휇31 휇20 휇21 휇26 휇27 휇16 휇17 휇22 휇23 휇28 휇29 휇3 휇3 휇2 휇9 휇8 휇7 휇14 휇5 휇12 휇11 휇10 휇1 휇0 휇15 휇6 휇13 휇4 휇19 휇26 휇17 휇24 휇31 휇22 휇29 휇20 휇27 휇18 휇25 휇16 휇23 휇30 휇21 휇28 휇4 휇4 휇13 휇14 휇15 휇8 휇1 휇2 휇3 휇12 휇5 휇6 휇7 휇0 휇9 휇10 휇11 휇20 휇21 휇22 휇23 휇24 휇25 휇26 휇27 휇28 휇29 휇30 휇31 휇16 휇17 휇18 휇19 휇5 휇5 휇4 휇15 휇6 휇9 휇8 휇11 휇2 휇13 휇12 휇7 휇14 휇1 휇0 휇3 휇10 휇21 휇28 휇23 휇30 휇17 휇24 휇19 휇26 휇29 휇20 휇31 휇22 휇25 휇16 휇27 휇18 휇6 휇6 휇7 휇4 휇13 휇10 휇3 휇8 휇9 휇14 휇15 휇12 휇5 휇2 휇11 휇0 휇1 휇22 휇31 휇28 휇21 휇18 휇27 휇24 휇17 휇30 휇23 휇20 휇29 휇26 휇19 휇16 휇25 휇7 휇7 휇14 휇5 휇4 휇11 휇10 휇1 휇8 휇15 휇6 휇13 휇12 휇3 휇2 휇9 휇0 휇23 휇22 휇29 휇28 휇19 휇18 휇25 휇24 휇31 휇30 휇21 휇20 휇27 휇26 휇17 휇16 휇8 휇8 휇9 휇10 휇11 휇12 휇13 휇14 휇15 휇8 휇1 휇2 휇3 휇4 휇5 휇6 휇7 휇24 휇25 휇26 휇27 휇28 휇29 휇30 휇31 휇16 휇17 휇18 휇19 휇20 휇21 휇22 휇23 휇9 휇9 휇0 휇11 휇2 휇13 휇4 휇7 휇14 휇1 휇8 휇3 휇10 휇5 휇12 휇15 휇6 휇25 휇16 휇19 휇26 휇21 휇28 휇31 휇22 휇17 휇24 휇27 휇18 휇29 휇20 휇23 휇30 휇10 휇10 휇3 휇0 휇9 휇14 휇15 휇4 휇5 휇2 휇11 휇8 휇1 휇6 휇7 휇12 휇13 휇26 휇27 휇16 휇17 휇22 휇23 휇28 휇29 휇18 휇19 휇24 휇25 휇30 휇31 휇20 휇21 휇11 휇11 휇10 휇1 휇0 휇15 휇6 휇13 휇4 휇3 휇2 휇9 휇8 휇7 휇14 휇5 휇12 휇27 휇18 휇25 휇16 휇23 휇30 휇21 휇28 휇19 휇26 휇17 휇24 휇31 휇22 휇29 휇20 휇12 휇12 휇5 휇6 휇7 휇0 휇9 휇10 휇11 휇4 휇13 휇14 휇15 휇8 휇1 휇2 휇3 휇28 휇29 휇30 휇31 휇16 휇17 휇18 휇19 휇20 휇21 휇22 휇23 휇24 휇25 휇26 휇27 휇13 휇13 휇12 휇7 휇14 휇1 휇0 휇3 휇10 휇5 휇4 휇15 휇6 휇9 휇8 휇11 휇2 휇29 휇20 휇31 휇22 휇25 휇16 휇27 휇18 휇21 휇28 휇23 휇30 휇17 휇24 휇19 휇26 휇14 휇14 휇15 휇12 휇5 휇2 휇11 휇0 휇1 휇6 휇7 휇4 휇13 휇10 휇3 휇8 휇9 휇30 휇23 휇20 휇29 휇26 휇19 휇16 휇25 휇22 휇31 휇28 휇21 휇18 휇27 휇24 휇17 휇15 휇15 휇6 휇13 휇12 휇3 휇2 휇9 휇0 휇7 휇14 휇5 휇4 휇11 휇10 휇1 휇8 휇31 휇30 휇21 휇20 휇27 휇26 휇17 휇16 휇23 휇22 휇29 휇28 휇19 휇18 휇25 휇24 휇16 휇16 휇25 휇26 휇27 휇28 휇29 휇30 휇31 휇24 휇17 휇18 휇19 휇20 휇21 휇22 휇23 휇8 휇1 휇2 휇3 휇4 휇5 휇6 휇7 휇0 휇9 휇10 휇11 휇12 휇13 휇14 휇15 휇17 휇17 휇16 휇27 휇18 휇29 휇20 휇23 휇30 휇25 휇24 휇19 휇26 휇21 휇28 휇31 휇22 휇9 휇8 휇3 휇10 휇5 휇12 휇15 휇6 휇1 휇0 휇11 휇2 휇13 휇4 휇7 휇14 휇18 휇18 휇19 휇16 휇25 휇30 휇31 휇20 휇21 휇26 휇27 휇24 휇17 휇22 휇23 휇28 휇29 휇10 휇11 휇8 휇1 휇6 휇7 휇12 휇13 휇2 휇3 휇0 휇9 휇14 휇15 휇4 휇5 휇19 휇19 휇26 휇17 휇16 휇31 휇22 휇29 휇20 휇27 휇18 휇25 휇24 휇23 휇30 휇21 휇28 휇11 휇2 휇9 휇8 휇7 휇14 휇5 휇12 휇3 휇10 휇1 휇0 휇15 휇6 휇13 휇4 휇20 휇20 휇21 휇22 휇23 휇16 휇25 휇26 휇27 휇28 휇29 휇30 휇31 휇24 휇17 휇18 휇19 휇12 휇13 휇14 휇15 휇8 휇1 휇2 휇3 휇4 휇5 휇6 휇7 휇0 휇9 휇10 휇11 휇21 휇21 휇28 휇23 휇30 휇17 휇16 휇19 휇26 휇29 휇20 휇31 휇22 휇25 휇24 휇27 휇18 휇13 휇4 휇15 휇6 휇9 휇8 휇11 휇2 휇5 휇12 휇7 휇14 휇1 휇0 휇3 휇10 휇22 휇22 휇31 휇28 휇21 휇18 휇27 휇16 휇17 휇30 휇23 휇20 휇29 휇26 휇19 휇24 휇25 휇14 휇7 휇4 휇13 휇10 휇3 휇8 휇9 휇6 휇15 휇12 휇5 휇2 휇11 휇0 휇1 휇23 휇23 휇22 휇29 휇28 휇19 휇18 휇25 휇16 휇31 휇30 휇21 휇20 휇27 휇26 휇17 휇24 휇15 휇14 휇5 휇4 휇11 휇10 휇1 휇8 휇7 휇6 휇13 휇12 휇3 휇2 휇9 휇0 휇24 휇24 휇17 휇18 휇19 휇20 휇21 휇22 휇23 휇16 휇25 휇26 휇27 휇28 휇29 휇30 휇31 휇0 휇9 휇10 휇11 휇12 휇13 휇14 휇15 휇8 휇1 휇2 휇3 휇4 휇5 휇6 휇7 휇25 휇25 휇24 휇19 휇26 휇21 휇28 휇31 휇22 휇17 휇16 휇27 휇18 휇29 휇20 휇23 휇30 휇1 휇0 휇11 휇2 휇13 휇4 휇7 휇14 휇9 휇8 휇3 휇10 휇5 휇12 휇15 휇6 휇26 휇26 휇27 휇24 휇17 휇22 휇23 휇28 휇29 휇18 휇19 휇16 휇25 휇30 휇31 휇20 휇21 휇2 휇3 휇0 휇9 휇14 휇15 휇4 휇5 휇10 휇11 휇8 휇1 휇6 휇7 휇12 휇13 휇27 휇27 휇18 휇25 휇24 휇23 휇30 휇21 휇28 휇19 휇26 휇17 휇16 휇31 휇22 휇29 휇20 휇3 휇10 휇1 휇0 휇15 휇6 휇13 휇4 휇11 휇2 휇9 휇8 휇7 휇14 휇5 휇12 휇28 휇28 휇29 휇30 휇31 휇24 휇17 휇18 휇19 휇20 휇21 휇22 휇23 휇16 휇25 휇26 휇27 휇4 휇5 휇6 휇7 휇0 휇9 휇10 휇11 휇12 휇13 휇14 휇15 휇8 휇1 휇2 휇3 휇29 휇29 휇20 휇31 휇22 휇25 휇24 휇27 휇18 휇21 휇28 휇23 휇30 휇17 휇16 휇19 휇26 휇5 휇12 휇7 휇14 휇1 휇0 휇3 휇10 휇13 휇4 휇15 휇6 휇9 휇8 휇11 휇2 휇30 휇30 휇23 휇20 휇29 휇26 휇19 휇24 휇25 휇22 휇31 휇28 휇21 휇18 휇27 휇16 휇17 휇6 휇15 휇12 휇5 휇2 휇11 휇0 휇1 휇14 휇7 휇4 휇13 휇10 휇3 휇8 휇9 휇31 휇31 휇30 휇21 휇20 휇27 휇26 휇17 휇24 휇23 휇22 휇29 휇28 휇19 휇18 휇25 휇16 휇7 휇6 휇13 휇12 휇3 휇2 휇9 휇0 휇15 휇14 휇5 휇4 휇11 휇10 휇1 휇8

Observations

1. Every element appears only once in every row and every column. Thus every two elements

define a third one uniquely i.e 𝕆 ⋊ 푆° is a quasigroup.

2. 휇0 is the two sided identity of 𝕆 ⋊ 푠° i.e 휇0. 휇푖 = 휇푖. 휇0 = 휇푖 , for all 푖 ∈ {0,1, … ,31}. We

conclude that Octonion split extension forms a loop.

3. 𝕆 ⋊ 푆° is not associative i.e (휇푖휇푗 )휇푘 ≠ 휇푖(휇푗휇푘) ∀푖

Example 3.3.1

휇7(휇10휇14) = 휇7. 휇12 = 휇3 ≠ (휇7휇10)휇14 = 휇13. 휇14 = 휇11

(푒7, 1){(−푒2, 1)(−푒6, 1)} = (푒7, 1)(푒4, 1) = (푒3, 1) ≠ {(푒7, 1)(−푒2, 1)}(−푒6, 1)

= (−푒5, 1)(−푒6, 1) = (−푒3, 1)

3.3.1 Cyclic subloops of octonion split extensions

The cyclic subloop generated by 휇1 is denoted by 〈휇1〉 and is given by:

〈휇1〉 = {휇1, 휇8, 휇9, 휇0} = 〈휇9〉

It is constructed as follows:

1 1 휇1 = (푒1, 1) = (푒1, 1) = 휇1

2 휇1 = (푒1, 1)(푒1, 1) = (−푒0, 1) = 휇8

3 ( ) 휇1 = (−푒0, 1)(푒1, 1)= −푒1, 1 = 휇9

4 휇1 = (−푒1, 1)(푒1, 1) = (푒0, 1) = 휇0

Similarly, other cyclic subloops generated by other elements are:

1. 〈휇2〉 = {휇2, 휇8, 휇10, 휇0} = 〈휇10〉

2. 〈휇3〉 = {휇3, 휇8, 휇11, 휇0} = 〈휇11〉

3. 〈휇4〉 = {휇4, 휇8, 휇12, 휇0} = 〈휇12〉

4. 〈휇5〉 = {휇5, 휇8, 휇13, 휇0} = 〈휇13〉

5. 〈휇6〉 = {휇6, 휇8, 휇14, 휇0} = 〈휇14〉

6. 〈휇7〉 = {휇7, 휇8, 휇15, 휇0} = 〈휇15〉

7. 〈휇8〉 = {휇8, 휇0}

8. 〈휇17〉 = {휇17, 휇8, 휇25, 휇0} = 〈휇25〉

9. 〈휇18〉 = {휇18, 휇8, 휇26, 휇0} = 〈휇26〉

10. 〈휇19〉 = {휇19, 휇8, 휇27, 휇0} = 〈휇27〉

Remark 3.3.1.1 In the octonion split extension loop, the commutant is 〈휇8〉 = {휇8, 휇0}, and |〈휇8〉| = 2.

Theorem 3.3.1 Let 〈휇푖〉 be a cyclic subloop of 𝕆 ⋊ 푆°, then:

1. The elements of the cyclic subloop 〈휇푖〉 can be generated using Nim addition as follows:

〈휇푖〉 = {휇푖, 휇푐, 휇푐⊕ 푖, 휇0}

where, 휇푖 is the generating element, for all 푖 ∈ {0,1, … ,31} and 휇푐 is the non-identity element in the commutant, 퐶(𝕆 ⋊ 푆°).

For example,

〈휇25〉 = {휇25, 휇8, 휇25 ⨁ 8, 휇0} = {휇25, 휇8, 휇17, 휇0}

2. 〈휇푖〉 = 〈휇푖 ⨁ 푐〉 ∀ 휇푖 ∈ 𝕆 ⋊ 푆° and 푐 the subscript for the non-identity element in

퐶(𝕆 ⋊ 푆°) provided that 푖 ≠ 0.

3.3.2 Coset decomposition

We now construct the distinct cosets using some cyclic subloops of the octonion split extensions.

Examples 3.3.2.1

1. Let S = 〈휇1〉 = {휇1, 휇8, 휇9, 휇0} = 〈휇9〉 then,

휇2푆 = 휇2{휇1, 휇8, 휇9, 휇0} = {휇11, 휇10, 휇3, 휇2},

휇4푆 = 휇4{휇1, 휇8, 휇9, 휇0} = {휇13, 휇12, 휇5, 휇4},

휇6푆 = 휇6{휇1, 휇8, 휇9, 휇0} = {휇7, 휇14, 휇15, 휇6},

휇16푆 = 휇16{휇1, 휇8, 휇9, 휇0} = {휇25, 휇24, 휇17, 휇16},

휇18푆 = 휇18{휇1, 휇8, 휇9, 휇0} = {휇19, 휇26, 휇27, 휇18},

휇20푆 = 휇20{휇1, 휇8, 휇9, 휇0} = {휇21, 휇28, 휇29, 휇20},

휇22푆 = 휇22{휇1, 휇8, 휇9, 휇0} = {휇31, 휇30, 휇23, 휇22}

푆휇2 = {휇1, 휇8, 휇9, 휇0}휇2 = {휇3, 휇10, 휇11, 휇2},

푆휇4 = {휇1, 휇8, 휇9, 휇0}휇4 = {휇5, 휇12, 휇13, 휇4},

푆휇6 = {휇1, 휇8, 휇9, 휇0}휇6 = {휇15, 휇14, 휇7, 휇6},

푆휇16 = {휇1, 휇8, 휇9, 휇0}휇16 = {휇17, 휇24, 휇25, 휇16},

푆휇18 = {휇1, 휇8, 휇9, 휇0}휇18 = {휇27, 휇26, 휇19, 휇18},

푆휇20 = {휇1, 휇8, 휇9, 휇0}휇20 = {휇29, 휇28, 휇21, 휇20},

푆휇22 = {휇1, 휇8, 휇9, 휇0}휇22 = {휇23, 휇30, 휇31, 휇22}

2. For S= 〈휇3〉 = {휇3, 휇8, 휇11, 휇0} = 〈휇11〉 then,

휇1푆 = 휇1{휇3, 휇8, 휇11, 휇0} = {휇10, 휇9, 휇2, 휇1},

휇4푆 = 휇4{휇3, 휇8, 휇11, 휇0} = {휇15, 휇12, 휇7, 휇4},

휇5푆 = 휇5{휇3, 휇8, 휇11, 휇0} = {휇6, 휇13, 휇14, 휇5},

휇16푆 = 휇16{휇3, 휇8, 휇11, 휇0} = {휇27, 휇24, 휇19, 휇16},

휇17푆 = 휇17{휇3, 휇8, 휇11, 휇0} = {휇18, 휇25, 휇26, 휇17},

휇20푆 = 휇20{휇3, 휇8, 휇11, 휇0} = {휇23, 휇28, 휇31, 휇20},

휇22푆 = 휇22{휇3, 휇8, 휇11, 휇0} = {휇21, 휇30, 휇29, 휇22}

푆휇1 = {휇3, 휇8, 휇11, 휇0}휇1 = {휇2, 휇9, 휇10, 휇1},

푆휇4 = {휇3, 휇8, 휇11, 휇0}휇4 = {휇7, 휇12, 휇15, 휇4},

푆휇5 = {휇3, 휇8, 휇11, 휇0}휇5 = {휇14, 휇13, 휇6, 휇5},

푆휇16 = {휇3, 휇8, 휇11, 휇0}휇16 = {휇19, 휇24, 휇27, 휇16},

푆휇17 = {휇3, 휇8, 휇11, 휇0}휇17 = {휇26, 휇25, 휇18, 휇17},

푆휇20 = {휇3, 휇8, 휇11, 휇0}휇20 = {휇31, 휇28, 휇23, 휇20},

푆휇22 = {휇3, 휇8, 휇11, 휇0}휇22 = {휇29, 휇30, 휇21, 휇22}

In a similar way, the left and right cosets of the other cyclic subloops of the octonion split extensions can be obtained.

Theorem 3.3.2 Let 푆 = 〈휇푖〉 = {휇푖, 휇푐, 휇(푖 ⊕ 푐), 휇0}, be a cyclic subloop of 𝕆 ⋊ 푆°, the left cosets of 푆 can be obtained by Nim addition as follows:

휇푗 푆 = {휇푗 ⨁ 푖, 휇푗 ⨁ 푐, 휇푗⨁(푖 ⨁ 푐), 휇푗 ⨁ 0}

where 휇푗 ∈ 𝕆 ⋊ 푆° and 휇푗 ∉ 푆 ∀푖, 푗 ∈ {0, 1, 2, … ,31}. On the other hand, the right cosets of 푆 are obtained as follows:

푆휇푗 = {휇푖 ⨁ 푗, 휇푐 ⨁ 푗, 휇(푖 ⨁ 푐)⊕푗, 휇0 ⨁ 푗}

Example 3.3.2.2

If 푆 = 〈휇3〉 = {휇3, 휇8, 휇11, 휇0}, then the subscripts for the elements of 휇12푆 are obtained by:

휇12푆 = 휇12{휇3, 휇8, 휇11, 휇0}

= {휇12휇3, 휇12휇8, 휇12휇11, 휇12휇0}

= {휇12⨁3, 휇12⨁8, 휇12⨁11, 휇12⨁0}

= {휇15, 휇4, 휇7, 휇12}.

Conclusions

Given ∀ 휇푖, 휇푗 ∈ 𝕆 ⋊ 푆°, 푖, 푗 ∈ {0, 1, 2, … ,31}

(i) 휇푖 ∈ 휇푖푆 or 휇푖 ∈ 푆휇푖

(ii) 휇푖푆 = 휇푗푆 and 휇푖푆 ∩ 휇푗푆 = ∅ or 푆휇푖 = 푆휇푗 and 푆휇푖 ∩ 푆휇푗 = ∅

(iii) | 휇푖푆| = | 휇푗푆| or | 푆휇푖 | = | 푆휇푗| .

Therefore, the left or right cosets of a cyclic subloop 푆 partition 𝕆 ⋊ 푆° into equivalence classes under the relation 휇푖~휇푗.

3. 4 Sedenion split extension

This is the algebra that immediately follows the octonions. The sedenions form a 24-dimensional non-commutative and non –associative algebra over the reals. They retain the algebraic property of power associativity but lose the property of being an alternative algebra.

The basis elements for the sedenions are:

{푏0 = (푒0, 0), 푏1 = (푒1, 0), … , 푏7 = (푒7, 0), 푏8 = (0,1), 푏9 = (0, 푒1), … , 푏15 = (0, 푒7)}

The sedenion split extension 𝕊 ⋊ 푆° form a set of order 64. Its elements are of the form (푥, 푦) where 푥 ∈ 𝕊 and 푦 ∈ 푆0 and are defined as follows:

휃0 = (푏0, 1), 휃1 = (푏1, 1), 휃2 = (푏2, 1), 휃3 = (푏3, 1), 휃4 = (푏4, 1),

휃5 = (푏5, 1), 휃6 = (푏6, 1), 휃7 = (푏7, 1), 휃8 = (푏8, 1), 휃9 = (푏9, 1),

휃10 = (푏10, 1), 휃11 = (푏11, 1), 휃12 = (푏12, 1), 휃13 = (푏13, 1), 휃14 = (푏14, 1),

휃15 = (푏15, 1), 휃16 = (−푏0, 1), 휃17 = (−푏1, 1), 휃18 = (−푏2, 1), 휃19 = (−푏3, 1),

휃20 = (−푏4, 1), 휃21 = (−푏5, 1), 휃22 = (−푏6, 1), 휃23 = (−푏7, 1), 휃24 = (−푏8, 1),

휃25 = (−푏9, 1), 휃26 = (−푏10, 1), 휃27 = (−푏11, 1), 휃28 = (−푏12, 1), 휃29 = (−푏13, 1),

휃30 = (−푏14, 1), 휃31 = (−푏15, 1), 휃32 = (−푏0, 1), 휃33 = (푏1, −1), 휃34 = (푏2, −1),

휃35 = (푏3, −1), 휃36 = (푏4, −1), 휃37 = (푏5, −1), 휃38 = (푏6, −1), 휃39 = (푏7, −1),

휃40 = (푏8, −1), 휃41 = (푏9, −1), 휃42 = (푏10, −1), 휃43 = (푏11, −1), 휃44 = (푏12, −1),

휃45 = (푏13, −1), 휃46 = (푏14, −1), 휃47 = (푏15, −1), 휃48 = (−푏0, −1), 휃49 = (−푏1, −1),

휃50 = (−푏2, −1), 휃51 = (−푏3, −1), 휃52 = (−푏4, −1), 휃53 = (−푏5, −1), 휃54 = (−푏6, −1),

휃55 = (−푏7, −1), 휃56 = (−푏8, −1), 휃57 = (−푏9, −1), 휃58 = (−푏10, −1), 휃59 = (−푏11, −1),

휃60 = (−푏12, −1), 휃61 = (−푏13, −1), 휃62 = (−푏14, −1), 휃63 = (−푏15, −1)

The multiplication table for these elements can be obtained by using the Cayley-Dickson and the

Jonathan Smith doubling processes.

3.4.1 Cyclic subloops of sedenion split extensions

In this section, we start by constructing cyclic subloops generated by some elements of the sedenion split extensions. Nim addition is then used to generalize the construction of cyclic subloops.

1. The elements of the cyclic subloop 〈휃1〉 = 〈(푏1, 1)〉 are generated as follows:

1 휃1 = (푏1, 1) = 휃1

2 휃1 = (푏1, 1)(푏1, 1) = (푏1푏1, 1) = (−1,1) = 휃16

3 휃1 = (푏1, 1)(−1,1) = (−푏1, 1) = 휃17

4 휃1 = (푏1, 1)(−푏1, 1) = (푏0, 1) = 휃0

Thus,

〈휃1〉 ={휃1, 휃16, 휃17, 휃0}= 〈휃17〉

2. The elements of 〈휃19〉 = 〈(−푏3, 1)〉 are generated as follows:

1 휃19 = (−푏3, 1) = 휃19

2 휃19 = (−푏3, 1)(−푏3, 1) = (−푏3. −푏3, 1) = (−1,1) = 휃16

3 휃19 = (−푏3, 1)(−1,1) = (푏3, 1) = 휃3

4 휃19 = (−푏3, 1)(푏3, 1) = (푏0, 1) = 휃0

Thus,

〈휃19〉 ={휃19, 휃16, 휃3, 휃0}= 〈휃3〉

3. 〈휃39〉 = 〈(푏7, −1)〉 then,

1 휃39 = (푏7, −1) = 휃39

2 ̅̅̅ 휃39 = (푏7, −1)(푏7, −1) = (−푏7푏7 ,1) = (−1,1) = 휃16

3 휃39 = (푏7, −1)(−1,1) = (−푏7, −1) = 휃55

4 ̅̅̅̅̅ 휃39 = (푏7, −1)(−푏7, −1) = (−푏7. −푏7, 1) = (푏0, 1) = 휃0

〈휃39〉 ={휃39, 휃16, 휃55, 휃0}= 〈휃55〉

4. 〈휃61〉 = 〈(−푏13, −1)〉 then,

1 휃61 = (−푏13, −1) = 휃61

2 휃61 = (−푏13, −1)(−푏13, −1) = (−푏13. −푏13, −1) = (−1,1) = 휃16

3 휃61 = (−푏13 , −1)(−1,1) = (푏13 , −1) = 휃45

4 ̅̅̅̅ 휃61 = (−푏13, −1) (푏13 , −1) = (푏13. 푏13, 1) = (푏0, 1) = 휃0

〈휃61〉 = {휃61, 휃16, 휃45, 휃0} = 〈휃45〉

In a similar way, cyclic subloops generated by the other elements can be obtained.

Remark 3.4.1.1 In this case 〈휇16〉 = {휇16, 휇0} is the commutant of 𝕊 ⋊ 푆° with |〈휇16〉| = 2.

Theorem 3.4.1 Let 〈휃푖〉 be a cyclic subloop of 𝕊 ⋊ 푆°, then:

1. The elements of the cyclic subloop 〈휃푖〉 can be generated using Nim addition as follows:

〈휃푖〉 = {휃푖 , 휃푐, 휃(푖 ⨁ 푐), 휃0}

where, 휃푖 is the generating element and 휃푐 is the non-identity element in the commutant

퐶(𝕊 ⋊ 푆°).

2. 〈휃푖〉 = 〈휃푖 ⨁ 푐〉 ∀ 휃푖 ∈ 𝕊 ⋊ 푆° and 푐 the subscript for the non-identity element in 퐶(𝕊 ⋊ 푆°)

provided that 푖 ≠ 0.

3.4.2 Coset decomposition

We now carry out coset decomposition and show that any two left or right cosets of the sedenion split extensions are either disjoint or identical. We also give general way of constructing the distinct cosets of the cyclic subloops.

Let 푆 = 〈휃1〉 = {휃1, 휃16, 휃17, 휃0} then the left cosets of S are obtained a follows:

휃2푆 = 휃2{휃1, 휃16, 휃17, 휃0} = {휃19, 휃18, 휃3, 휃2}

휃4푆 = 휃4{휃1, 휃16, 휃17, 휃0} = {휃21, 휃20, 휃5, 휃4}

휃6푆 = 휃6{휃1, 휃16, 휃17, 휃0} = {휃7, 휃22, 휃23, 휃6}

휃8푆 = 휃8{휃1, 휃16, 휃17, 휃0} = {휃25, 휃24, 휃9, 휃8}

휃10푆 = 휃10{휃1, 휃16, 휃17, 휃0} = {휃11, 휃26, 휃27, 휃10}

휃12푆 = 휃12{휃1, 휃16, 휃17, 휃0} = {휃13, 휃28, 휃29, 휃12}

휃14푆 = 휃14{휃1, 휃16, 휃17, 휃0} = {휃31, 휃30, 휃15, 휃14}

휃32푆 = 휃32{휃1, 휃16, 휃17, 휃0} = {휃49, 휃48, 휃33, 휃32}

휃34푆 = 휃34{휃1, 휃16, 휃17, 휃0} = {휃35, 휃50, 휃51, 휃34}

휃36푆 = 휃36{휃1, 휃16, 휃17, 휃0} = {휃37, 휃52, 휃53, 휃36}

휃38푆 = 휃38{휃1, 휃16, 휃17, 휃0} = {휃55, 휃54, 휃39, 휃38}

휃40푆 = 휃40{휃1, 휃16, 휃17, 휃0} = {휃41, 휃56, 휃57, 휃40}

휃42푆 = 휃42{휃1, 휃16, 휃17, 휃0} = {휃59, 휃58, 휃43, 휃42}

휃44푆 = 휃44{휃1, 휃16, 휃17, 휃0} = {휃61, 휃60, 휃45, 휃44}

휃46푆 = 휃46{휃1, 휃16, 휃17, 휃0} = {휃47, 휃62, 휃63, 휃46}

Similarly, the left and right cosets of other cyclic subloops of the sedenion split extensions can be obtained.

Theorem 3.4.2 Let 푆 = {휃푖 , 휃푐, 휃(푖 ⨁ 푐), 휃0}, be a cyclic subloop of 𝕊 ⋊ 푆°, then the left cosets of 푆 can be obtained by Nim addition as follows:

휃푗푆 = {휃푗 ⨁ 푖, 휃푗 ⨁ 푐, 휃푗⨁(푖 ⨁ 푐), 휃푗⨂0}

Similarly, the right cosets can be obtained as follows:

푆휃푗 = {휃푖 ⨁ 푗, 휃푐 ⨁ 푗, 휃(푖 ⨁ 푐)⨁ 푗, 휃푗}, ∀푖, 푗 ∈ {0,1, 2, … ,63}.

where 휃푗 ∈ 𝕊 ⋊ 푆°.

Conclusions

Given ∀ 휃푖, 휃푗 ∈ 𝕊 ⋊ 푆°, 푖, 푗 ∈ {0,1,2, … ,63}

(i) 휃푖 ∈ 휃푖푆 or 휃푖 ∈ 푆휃푖

(ii) 휃푖푆 = 휃푗푆 and 휃푖푆 ∩ 휃푗푆 = ∅ or 푆휃푖 = 푆휃푗 and 푆휃푖 ∩ 푆휃푗 = ∅

(iii) |휃푖푆| = |휃푗푆| or |푆휃푖| = |푆휃푗 | .

Therefore, the left or right cosets of a cyclic subloop 푆 partition 𝕊 ⋊ 푆° into equivalence classes under the relation 휃푖~휃푗

CHAPTER FOUR: QUOTIENT LOOPS

4.0 Introduction

In Group Theory, if G is a group and H its subgroup such that 푔퐻 = 퐻푔 for all 푔 ∈ 퐺, then the set

퐺 ∕ 퐻 = {푔퐻, 푔 ∈ 퐺} of all cosets of 퐻 in 퐺 is called a quotient or factor group with the operation defined by (푎퐻) ∗ (푏퐻) = 푎푏퐻 ∀ 푎, 푏 ∈ 퐺. We use this definition to construct quotient loops using cyclic normal subloops of order two and four. In each case, we show that the multiplication of the elements of the quotient loop formed can also be carried out using Nim-addition.

4.1 Quotient loops arising from complex split extensions

First, we consider a cyclic normal subloop of order 2 i.e. 푆 = 〈훼2〉 = {훼2, 훼0}. The left cosets of

푆 are 훼1푆 = {훼3, 훼1 }, 훼4푆 = {훼6, 훼4 } and 훼5푆 = {훼7, 훼5}.

The elements of the quotient loop ℂ ⋊ 푆°/푆 are 훼0푆, 훼1푆, 훼4푆, and 훼5푆 and the multiplication of these elements is generally given by 훼푖푆훼푗푆 = 훼푖훼푗푆. The following table gives the multiplication for the elements of ℂ ⋊ 푆°/푆.

Table 4.1.1 Multiplication of the elements of ℂ ⋊ 푆°/푆

훼0푆 훼1푆 훼4푆 훼5푆 훼0푆 훼0푆 훼1푆 훼4푆 훼5푆 훼1푆 훼1푆 훼0푆 훼5푆 훼4푆 훼4푆 훼4푆 훼5푆 훼0푆 훼1푆 훼5푆 훼5푆 훼4푆 훼1푆 훼0푆

To construct the Nim addition table of the subscripts of the elements of ℂ ⋊ 푆°/푆, refer Table 1.1.

For clarity, when comparing the table above and the Nim addition table, it is important to note that, 훼0푆 = 훼2푆, 훼1푆 = 훼3푆, 훼4푆 = 훼6푆, and 훼5푆 = 훼7푆.

Next, we consider a cyclic normal subloop of order 4. For example, let

푀 = 〈훼5〉 = {훼5 , 훼2 , 훼7 , 훼0} = 〈훼7〉. The distinct left cosets of 푀 are 훼0푀 = {훼5 , 훼2 , 훼7 , 훼0} and 훼1푀 = {훼6, 훼3, 훼4, 훼0}.

Therefore, the elements of the quotient loop ℂ ⋊ 푆°/푀 are 훼0푀 and 훼1푀. The multiplication of these elements is given by the following table.

Table 4.1.2 Multiplication of the elements of ℂ ⋊ 푆°/푀

훼0푀 훼1푀 훼0푀 훼0푀 훼1푀 훼1푀 훼1푀 훼0푀

To construct the Nim addition table of the subscripts of the elements of ℂ ⋊ 푆°/푀, refer Table 1.1.

Observations

From the above tables we observe that ∀푖, 푗, 푘 ∈ {0,1, … ,7} and 푆 a cyclic subloop of ℂ ⋊ 푆°

(i) 훼푖푆. 훼푗푆 = 훼푖 ⨁ 푗푆

(ii) 훼푖푆. 훼푗푆 = 훼푗푆. 훼푖푆 since 푖 ⨁ 푗 = 푗 ⨁ 푖

(iii) 훼푖푆. 훼푖푆 = 훼0푆 since 푖 ⨁ 푖 = 0

(iv) 훼푖푆(훼푗푆훼푘푆) = (훼푖푆훼푗푆)훼푘푆 since 푖 ⨁ (푗 ⨁ 푘) = (푖 ⨁ 푗) ⨁ 푘

We conclude that the subscripts of the elements of a quotient loop formed from complex split extensions are closed under Nim addition. The multiplication of the elements of the quotient loop can therefore be carried out using Nim addition.

4.2 Quotient loops arising from quaternion split extension

In this section we construct quotient loops using cyclic normal subloops of the quaternion split extensions. First, we consider a cyclic subloop of order 2 given by 푆 = 〈훽8〉 = {훽8, 훽0}. The left cosets of 푆 are:

훽0푆 = 훽0{훽8, 훽0} = {훽8, 훽0}, 훽1푆 = 훽1{훽8, 훽0} = {훽9, 훽1},

훽2푆 = 훽2{훽8, 훽0} = {훽10, 훽2}, 훽3푆 = 훽3{훽8, 훽0} = {훽11, 훽3},

훽4푆 = 훽4{훽8, 훽0} = {훽12, 훽4}, 훽5푆 = 훽5{훽8, 훽0} = {훽13, 훽5},

훽6푆 = 훽6{훽8, 훽0} = {훽14, 훽6}, and 훽7푆 = 훽7{훽8, 훽0} = {훽15, 훽7}.

The elements of the quotient loop ℍ ⋊ 푆°/푆 are 훽0푆, 훽1푆, 훽2푆, 훽3푆, 훽4푆, 훽5푆, 훽6푆 and

훽7푆. The multiplication of these elements is given by the following table.

Table 4.2.1 Multiplication of the elements of ℍ ⋊ 푆°/푆

훽0푆 훽1푆 훽2푆 훽3푆 훽4푆 훽5푆 훽6푆 훽7푆 훽0푆 훽0푆 훽1푆 훽2푆 훽3푆 훽4푆 훽5푆 훽6푆 훽7푆 훽1푆 훽1푆 훽0푆 훽3푆 훽2푆 훽5푆 훽4푆 훽7푆 훽6푆 훽2푆 훽2푆 훽3푆 훽0푆 훽1푆 훽6푆 훽7푆 훽4푆 훽5푆 훽3푆 훽3푆 훽2푆 훽1푆 훽0푆 훽7푆 훽6푆 훽5푆 훽4푆 훽4푆 훽4푆 훽5푆 훽6푆 훽7푆 훽0푆 훽1푆 훽2푆 훽3푆 훽5푆 훽5푆 훽4푆 훽7푆 훽6푆 훽1푆 훽0푆 훽3푆 훽2푆 훽6푆 훽6푆 훽7푆 훽4푆 훽5푆 훽2푆 훽3푆 훽0푆 훽1푆 훽7푆 훽7푆 훽6푆 훽5푆 훽4푆 훽3푆 훽2푆 훽1푆 훽0푆

To construct the Nim addition table of the subscripts of the elements of ℍ ⋊ 푆°/푆 i.e. 0, 1, 2, 3, 4,

5, 6 and 7 refer Table 1.1.

Next we construct a quotient loop using a cyclic normal subloop 푀 = 〈훽2〉 = {훽2, 훽8, 훽10, 훽0} of order 4. The left cosets of 푀 are:

훽0푀 = {훽2, 훽8, 훽10, 훽0}, 훽1푀 = {훽3, 훽9, 훽11, 훽1},

훽4푀 = {훽14, 훽12, 훽6, 훽4} and 훽5푀 = {훽15, 훽13, 훽7, 훽5}.

Therefore, the elements of the quotient loop ℍ ⋊ 푆°/푀 are, 훽0푀, 훽1푀, 훽4푀 and 훽5푀.

The multiplication of these elements is given by the following table.

Table 4.2.2 Multiplication of the elements of ℍ ⋊ 푆°/푀

훽0푀 훽1푀 훽4푀 훽5푀 훽0푀 훽0푀 훽1푀 훽4푀 훽5푀 훽1푀 훽1푀 훽0푀 훽5푀 훽4푀 훽4푀 훽4푀 훽5푀 훽0푀 훽1푀 훽5푀 훽5푀 훽4푀 훽1푀 훽0푀

To get the Nim addition table of elements 0, 1, 4 and 5 refer to Table 1.1.

From the above tables we observe that ∀푖, 푗, 푘 ∈ {0,1, … ,15} and 푆 a cyclic subloop of ℍ ⋊ 푆°

(i) 훽푖푆. 훽푗푆 = 훽푖 ⨁ 푗푆

(ii) 훽푖푆. 훽푗푆 = 훽푗푆. 훽푖푆 since 푖 ⨁ 푗 = 푗 ⨁ 푖

(iii) 훽푖푆. 훽푖푆 = 훽0푆 since 푖 ⨁ 푖 = 0

(iv) 훽푖푆(훽푗푆훽푘푆) = (훽푖푆훽푗푆)훽푘푆 since 푖 ⨁ (푗 ⨁ 푘) = (푖 ⨁ 푗) ⨁ 푘

For a subloop S of order 2 the number of distinct cosets is given by; 16 = 8 . 2

We again notice that the subscripts of the elements of a quotient loop formed from quaternion split extensions are closed under Nim addition. The multiplication of the elements of ℍ ⋊ 푆°/푆 can therefore be achieved by using Nim addition.

4.3 Quotient loops arising from octonion split extensions

If 푆 = 〈휇8〉 = {휇8, 휇0} is a cyclic normal subloop of 𝕆 ⋊ 푆° of order 2, then the left cosets of 푆 are:

휇0푆 = 휇0{휇8, 휇0} = {휇8, 휇0}, 휇1푆 = 휇1{휇8, 휇0} = {휇9, 휇1},

휇2푆 = 휇2{휇8, 휇0} = {휇10, 휇2}, 휇3푆 = 휇3{휇8, 휇0} = {휇11, 휇3},

휇4푆 = 휇4{휇8, 휇0} = {휇12, 휇4}, 휇5푆 = 휇5{휇8, 휇0} = {휇13, 휇5},

휇6푆 = 휇6{휇8, 휇0} = {휇14, 휇6}, 휇7푆 = 휇7{휇8, 휇0} = {휇15, 휇7},

휇16푆 = 휇16{휇8, 휇0} = {휇24, 휇16}, 휇17푆 = 휇17{휇8, 휇0} = {휇25, 휇17},

휇18푆 = 휇18{휇8, 휇0} = {휇26, 휇18}, 휇19푆 = 휇19{휇8, 휇0} = {휇27, 휇19},

휇20푆 = 휇20{휇8, 휇0} = {휇28, 휇20}, 휇21푆 = 휇21{휇8, 휇0} = {휇29, 휇21},

휇22푆 = 휇22{휇8, 휇0} = {휇30, 휇22}, and 휇23푆 = 휇23{휇8, 휇0} = {휇31, 휇23}

Therefore, the elements of 𝕆 ⋊ 푆° ∕ 푆 are:

휇0푆, 휇1푆, 휇2푆, 휇3푆, 휇4푆, 휇5푆, 휇6푆, 휇7푆, 휇16푆, 휇17푆, 휇18푆, 휇19푆, 휇20푆, 휇21푆,

휇22푆, and 휇23푆. The multiplication of these elements is given by the following table.

Table 4.3.1 Multiplication of the elements of 𝕆 ⋊ 푆°/푆

휇0푆 휇1푆 휇2푆 휇3푆 휇4푆 휇5푆 휇6푆 휇7푆 휇16푆 휇17푆 휇18푆 휇19푆 휇20푆 휇21푆 휇22푆 휇23푆 휇0푆 휇0푆 휇1푆 휇2푆 휇3푆 휇4푆 휇5푆 휇6푆 휇7푆 휇16푆 휇17푆 휇18푆 휇19푆 휇20푆 휇21푆 휇22푆 휇23푆

휇1푆 휇1푆 휇0푆 휇3푆 휇2푆 휇5푆 휇4푆 휇7푆 휇6푆 휇17푆 휇16푆 휇19푆 휇18푆 휇21푆 휇20푆 휇23푆 휇22푆 휇2푆 휇2푆 휇3푆 휇0푆 휇1푆 휇6푆 휇7푆 휇4푆 휇5푆 휇18푆 휇19푆 휇16푆 휇17푆 휇22푆 휇23푆 휇20푆 휇21푆

휇3푆 휇3푆 휇2푆 휇1푆 휇0푆 휇7푆 휇6푆 휇5푆 휇4푆 휇19푆 휇18푆 휇17푆 휇16푆 휇23푆 휇22푆 휇21푆 휇20푆 휇4푆 휇4푆 휇5푆 휇6푆 휇7푆 휇0푆 휇1푆 휇2푆 휇3푆 휇20푆 휇21푆 휇22푆 휇23푆 휇16푆 휇17푆 휇18푆 휇19푆 휇5푆 휇5푆 휇4푆 휇7푆 휇6푆 휇1푆 휇0푆 휇3푆 휇2푆 휇21푆 휇20푆 휇23푆 휇22푆 휇17푆 휇16푆 휇19푆 휇18푆

휇6푆 휇6푆 휇7푆 휇4푆 휇5푆 휇2푆 휇3푆 휇0푆 휇1푆 휇22푆 휇23푆 휇20푆 휇21푆 휇18푆 휇19푆 휇16푆 휇17푆 휇7푆 휇7푆 휇6푆 휇5푆 휇4푆 휇3푆 휇2푆 휇1푆 휇0푆 휇23푆 휇22푆 휇21푆 휇20푆 휇19푆 휇18푆 휇17푆 휇16푆

휇16푆 휇16푆 휇17푆 휇18푆 휇19푆 휇20푆 휇21푆 휇22푆 휇23푆 휇0푆 휇1푆 휇2푆 휇3푆 휇4푆 휇5푆 휇6푆 휇7푆 휇17푆 휇17푆 휇16푆 휇19푆 휇18푆 휇21푆 휇20푆 휇23푆 휇22푆 휇1푆 휇0푆 휇3푆 휇2푆 휇5푆 휇4푆 휇7푆 휇6푆 휇18푆 휇18푆 휇19푆 휇16푆 휇17푆 휇22푆 휇23푆 휇20푆 휇21푆 휇2푆 휇3푆 휇0푆 휇1푆 휇6푆 휇7푆 휇4푆 휇5푆

휇19푆 휇19푆 휇18푆 휇17푆 휇16푆 휇23푆 휇22푆 휇21푆 휇20푆 휇3푆 휇2푆 휇1푆 휇0푆 휇7푆 휇6푆 휇5푆 휇4푆 휇20푆 휇20푆 휇21푆 휇22푆 휇23푆 휇16푆 휇17푆 휇18푆 휇19푆 휇4푆 휇5푆 휇6푆 휇7푆 휇0푆 휇1푆 휇2푆 휇3푆 휇21푆 휇21푆 휇20푆 휇23푆 휇22푆 휇17푆 휇16푆 휇19푆 휇18푆 휇5푆 휇4푆 휇7푆 휇6푆 휇1푆 휇0푆 휇3푆 휇2푆

휇22푆 휇22푆 휇23푆 휇20푆 휇21푆 휇18푆 휇19푆 휇16푆 휇17푆 휇6푆 휇7푆 휇4푆 휇5푆 휇2푆 휇3푆 휇0푆 휇1푆 휇23푆 휇23푆 휇22푆 휇21푆 휇20푆 휇19푆 휇18푆 휇17푆 휇16푆 휇7푆 휇6푆 휇5푆 휇4푆 휇3푆 휇2푆 휇1푆 휇0푆

The Nim addition table of the elements 0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22 and 23 is given by Table 1.1.

Next, we let 푀 = 〈휇1〉 = {휇1, 휇8, 휇9, 휇0} = 〈휇9〉 be a cyclic normal subloop of order 4. The left cosets of 푀 are:

휇0푀 = {휇1, 휇8, 휇9, 휇0}, 휇2푀 = {휇11, 휇10, 휇3, 휇2},

휇4푀 = {휇13, 휇12, 휇5, 휇4}, 휇6푀 = {휇7, 휇14, 휇15, 휇6},

휇16푀 = {휇25, 휇24, 휇17, 휇16}, 휇18푀 = {휇19, 휇26, 휇27, 휇18},

휇20푀 = {휇21, 휇28, 휇29, 휇20}, and 휇22푀 = {휇31, 휇30, 휇23, 휇22}.

In this case, the elements of 𝕆 ⋊ 푆°/푀 are, 휇0푀, 휇2푀, 휇4푀, 휇6푀, 휇16푀, 휇18푀, 휇20푀, and

휇22푀. The multiplication of these elements is given by the following table.

Table 4.3.2 Multiplication of the elements of 𝕆 ⋊ 푆°/푀

휇0푀 휇2푀 휇4푀 휇6푀 휇16푀 휇18푀 휇20푀 휇22푀 휇0푀 휇0푀 휇2푀 휇4푀 휇6푀 휇16푀 휇18푀 휇20푀 휇22푀 휇2푀 휇2푀 휇0푀 휇6푀 휇4푀 휇18푀 휇16푀 휇22푀 휇20푀 휇4푀 휇4푀 휇6푀 휇0푀 휇2푀 휇20푀 휇22푀 휇16푀 휇18푀 휇6푀 휇6푀 휇4푀 휇2푀 휇0푀 휇22푀 휇20푀 휇18푀 휇16푀 휇16푀 휇16푀 휇18푀 휇20푀 휇22푀 휇0푀 휇2푀 휇4푀 휇6푀 휇18푀 휇18푀 휇16푀 휇22푀 휇20푀 휇2푀 휇0푀 휇6푀 휇4푀 휇20푀 휇20푀 휇22푀 휇16푀 휇18푀 휇4푀 휇6푀 휇0푀 휇2푀 휇22푀 휇22푀 휇20푀 휇18푀 휇16푀 휇6푀 휇4푀 휇2푀 휇0푀

To get the Nim addition table for the elements 0, 2, 4, 6, 16, 18, 20 and 22 refer Table 1.1.

Observations

From the above tables we observe that ∀ 푖, 푗, 푘 ∈ {0,1, … ,31} and for 푆 a subloop of 𝕆 ⋊ 푆°

(i) 휇푖푆. 휇푗푆 = 휇푖 ⨁ 푗푆

(ii) 휇푖푆. 휇푗푆 = 휇푗푆. 휇푖 푆 since 푖 ⨁ 푗 = 푗 ⨁ 푖

(iii) 휇푖푆. 휇푖푆 = 휇0푆 since 푖 ⨁ 푖 = 0

(iv) 휇푖푆(휇푗푆휇푘푆) = (휇푖푆휇푗푆)휇푘푆 since 푖⨁ (푗 ⨁ 푘) = (푖 ⨁ 푗) ⨁ 푘

For a subloop S of order 2 the number of distinct cosets is given by; 32 = 16 . 2

We conclude that the subscripts of the elements of the quotient loop generated from octonion split extension are closed under Nim addition. The multiplication of the elements of the quotient loop can thus be achieved by Nim addition.

4.4 Quotient loops arising from sedenion split extensions

The sedenion split extension form a 26-dimensional set. Therefore, a cyclic normal subloop of

5 order 2 will have 2 distinct cosets. If 푆 = 〈휃16〉 = {휃0, 휃16} , then the elements of the quotient loop 𝕊 ⋊ 푆°/푆 can be generated as follows:

휃0푆 = 휃0{휃0, 휃16} = {휃0, 휃16}, 휃1푆 = 휃1{휃0, 휃16} = {휃1, 휃17},

휃2푆 = 휃2{휃0, 휃16} = {휃2, 휃18}, 휃3푆 = 휃3{휃0, 휃16}, = {휃3, 휃19},

Continuing this way, the following other elements of 𝕊 ⋊ 푆°/푆 can be obtained,

휃4푆, 휃5푆, 휃6푆, 휃7푆, 휃8푆, 휃9푆, 휃10푆, 휃11푆, 휃12푆, 휃13푆,

휃14푆, 휃15푆, 휃32푆, 휃33푆, 휃34푆, 휃35푆, 휃36푆, 휃37푆, 휃38푆,

휃39푆, 휃40푆, 휃41푆, 휃42푆, 휃43푆, 휃44푆, 휃45푆, 휃46푆, and 휃47푆

The multiplication table for these elements can be achieved using the operation 휃푖푆휃푗푆 = 휃푖휃푗푆.

To get the Nim addition table for the subscripts of these elements, refer Table 1.1

On the other hand, if 푀 = 〈휃1〉 = {휃1, 휃16, 휃17, 휃0} is a cyclic normal subloop of order 4, then in this case, the quotient loop 𝕊 ⋊ 푆°/푀 will have the following 16 elements:

휃0푀, 휃2푀, 휃4푀, 휃6푀, 휃8푀, 휃10푀, 휃12푀, 휃14푀, 휃32푀

휃34푀, 휃36푀, 휃38푀, 휃40푀, 휃42푀, 휃44푀, and 휃46푀

The multiplication table for these elements can be constructed by using the operation

휃푖푀휃푗푀 = 휃푖휃푗푀. To construct the Nim addition table for subscripts of the elements of

𝕊 ⋊ 푆°/푀, refer Table 1.1.

Observations

We note that ∀ 푖, 푗, 푘 ∈ {0,1, … ,63}

(v) 휃푖푆. 휃푗푆 = 휃푖 ⨁ 푗푆

(vi) 휃푖푆. 휃푗푆 = 휃푗푆. 휃푖푆 since 푖 ⨁ 푗 = 푗 ⨁ 푖

(vii) 휃푖푆. 휃푖푆 = 휃0푆 since 푖 ⨁ 푖 = 0

(viii) 휃푖푆(휃푗푆. 휃푘푆) = (휃푖푆. 휃푗푆)휃푘푆 since 푖⨁ (푗 ⨁ 푘) = (푖 ⨁ 푗) ⨁ 푘

We conclude that the subscripts of these elements are closed under Nim addition of non-negative integers and that the multiplication of these elements can be done by just considering Nim addition of their subscripts.

4.5 Conclusions

In this chapter, quotient loops were constructed using cyclic normal subloops of order 2 and 4. The multiplication tables of the quotient loops were constructed and the results confirmed that the multiplication of the elements in the quotient loop is similar to just considering the Nim addition of the subscripts of the individual elements.

CHAPTER FIVE: CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER

RESEARCH

5.0 Introduction

In this chapter, we give conclusions and recommendations for further research.

5.1 Conclusions

The multiplication tables for the split extension of hypercomplex numbers were constructed by use of the Jonathan-Smith doubling formula. Cyclic subloops were then constructed. The results showed that the cyclic subloops of the split extension of hypercomplex numbers are either of order

2 or 4. Coset decomposition was carried out and the results also showed that the cosets of a cyclic subloop of a split extension loop form a partition of the loop i.e. any two left or right cosets of a cyclic subloop of a split extension loop are either disjoint or identical. Nim addition was also used to give a general way of generating cyclic subloops and distinct cosets arising from them. Finally, in chapter four, quotient loops were constructed and the results confirmed that the multiplication of the elements in the quotient loop is achievable by just considering the Nim-addition of the subscripts of the individual elements.

5.2 Recommendation for future research

This study mainly focused on determining whether cosets of cyclic subloops of the split extensions of hypercomplex numbers form a partition of the split extension loop. The question of whether the cosets of subloops that are not cyclic partition the loop still remains open.

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