MULTIPLICATION GROUPS AND INNER MAPPING GROUPS OF CAYLEY-DICKSON LOOPS

作者：Kirshtein Jenya^*

来源：Journal of Algebra and Its Applications, 2014, 13(1): 1350078.

DOI：10.1142/S0219498813500783

• Cayley-Dickson doubling process
• Cayley-Dickson loop
• multiplication group
• inner mapping group
• octonion
• sedenion

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(2n) 2 and describe the multiplication group Mlt(Q(n)) as a semidirect product of Inn(Q(n)) x Z(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(2n-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)) x Z(2) and an elementary abelian 2-group of order 2(n).

• 出版日期2014-2