Multiplication groups and inner mapping groups of Cayley–Dickson loops

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 2 −2 and describe the multiplication group Mlt(Qn) as a semidirect product of Inn(Qn) Z2 and an elementary abelian 2-group of order 2. We prove that one-sided inner mapping groups Innl(Qn) and Innr(Qn) are equal, elementary abelian 2-groups of order 22n−1−1. We establish that one-sided multiplication groups Mltl(Qn) and Mltr(Qn) are isomorphic, and show that Mltl(Qn) is a semidirect product of Innl(Qn) Z2 and an elementary abelian 2group of order 2.