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MULTIPLICATION GROUPS AND INNER MAPPING GROUPS OF CAYLEY–DICKSON LOOPS

JENYA KIRSHTEIN

Department of Mathematics, University of Denver, 2360 South Gaylord Street, Denver, CO 80208, USA

https://doi.org/10.1142/S0219498813500783Cited by:1 (Source: Crossref)

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 2^2ⁿ-2 and describe the multiplication group Mlt(Q_n) as a semidirect product of Inn(Q_n) × ℤ₂ and an elementary abelian 2-group of order 2ⁿ. 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) × ℤ₂ and an elementary abelian 2-group of order 2ⁿ.

Keywords:

AMSC: 20N05, 17D99

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