Multiplication groups and inner mapping groups of Cayley-Dickson loops

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.