2012/07/17 by Jenya Kirshtein, Kirshtein, Jenya
Mathematics · #17D99 #20N05 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR #msc:17D99 #msc:20N05
paper · pdf · doi:10.48550/arxiv.1207.4230
23 pages, 2 figures
arxiv created 2012/07/17 · arxiv updated 2012/07/19
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^(2n-2) and describe the multiplication group Mlt(Qn) as a semidirect product of Inn(Qn)xZ2 and an elementary abelian 2-group of order 2n. We prove that one-sided inner mapping groups Innl(Qn) and Innr(Qn) are equal, elementary abelian 2-groups of order 2^(2^(n-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)xZ2 and an elementary abelian 2-group of order 2n.