On Cosets In Split Extensions Of Hypercomplex Numbers

International Journal of Research and Innovation in Applied Science (IJRIAS) | Volume VI, Issue III, March 2021|ISSN 2454-6194

On Cosets In Split Extensions Of Hypercomplex Numbers

Musyoka David Mwanzia, Lydia Njuguna
Kenyatta University

ABSTRACT
This paper 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 S is a normal subloop of L will the left and right cosets of S coincide, these cosets form a loopL/Scalled the quotient or factor loop whose multiplication is defined by (a∙S)∙(b∙S)=(a∙b)∙S, ∀ a,b∈L. In this work we use cyclic normal subloops of split extensions of hypercomplex numbers to construct quotient loops, and 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. The complex split extension forms a group and hence it remains trivial to work on the same. Though the authors have also carried out the same process on the sedenion split extensions, the present paper focusesmainly on the quaternion and octonion split extension.
AMS Subject Classification: 20B05

Key words: Loop, Hypercomplex numbers, split extension, Cayley – Dickson, Quotient loop, Coset decomposition

I. INTRODUCTION

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 S 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 tothe question of when the cosets of a Steiner subloop Spartition 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 order2|S|, any two of which intersect in S.

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