MULTIPLICATION GROUPS AND INNER MAPPING GROUPS OF CAYLEY–DICKSON LOOPS

Abstract

The Cayley–Dickson loop Qn 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 (Qn) is an elementary abelian 2-group of order 22n-2 and describe the multiplication group Mlt (Qn) as a semidirect product of Inn (Qn) × ℤ2 and an elementary abelian 2-group of order 2n. We prove that one-sided inner mapping groups Inn l(Qn) and Inn r(Qn) are equal, elementary abelian 2-groups of order 22n-1-1. We establish that one-sided multiplication groups Mlt l(Qn) and Mlt r(Qn) are isomorphic, and show that Mlt l(Qn) is a semidirect product of Inn l(Qn) × ℤ2 and an elementary abelian 2-group of order 2n.