Multiplication groups and inner mapping groups of Cayley-Dickson loops

TL;DR

This paper investigates the algebraic structure of Cayley-Dickson loops, revealing detailed descriptions of their inner and multiplication groups, including their orders, isomorphisms, and decompositions, for all n.

Contribution

It provides explicit descriptions of the inner and multiplication groups of Cayley-Dickson loops, including their orders, structures, and relationships, extending understanding of these algebraic objects.

Findings

Inn(Q_n) is an elementary abelian 2-group of order 2^(2^n-2)

Mlt(Q_n) is a semidirect product involving Inn(Q_n) and an elementary abelian 2-group

One-sided inner and multiplication groups are equal or isomorphic, with specific structures

Abstract

The Cayley-Dickson loop Q_n is the multiplicative closure of basic elements of the algebra constructed by n applications of the Cayley-Dickson doubling process (the first few examples of such algebras are real numbers, complex numbers, quaternions, octonions, sedenions). We establish that the inner mapping group Inn(Q_n) is an elementary abelian 2-group of order 2^(2^n-2) and describe the multiplication group Mlt(Q_n) as a semidirect product of Inn(Q_n)xZ_2 and an elementary abelian 2-group of order 2^n. We prove that one-sided inner mapping groups Inn_l(Q_n) and Inn_r(Q_n) are equal, elementary abelian 2-groups of order 2^(2^(n-1)-1). We establish that one-sided multiplication groups Mlt_l(Q_n) and Mlt_r(Q_n) are isomorphic, and show that Mlt_l(Q_n) is a semidirect product of Inn_l(Q_n)xZ_2 and an elementary abelian 2-group of order 2^n.