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Automorphism groups of Cayley-Dickson loops. (English) Zbl 1258.20057

The Cayley-Dickson loops \((Q_n,\cdot)\) are defined inductively by \(Q_0=\{\pm 1\}\), \(Q_n=\{(x,0),(x,1)\mid x\in Q_{n-1}\}\) where \(Q_n\) is the multiplicative closure of basic elements of the algebra constructed by \(n\) applications of the Cayley-Dickson doubling process. Firstly, properties of the Cayley-Dickson loops are discussed. In particular, Cayley-Dickson loops are diassociative, inverse property loops and Hamiltonian. Secondly, the automorphism groups of Cayley-Dickson loops are described and the following theorem is established: For a Cayley-Dickson loop \(Q_n\) with \(n\geq 4\), the automorphism group \(\operatorname{Aut}(Q_n)\) is a direct product of \(\operatorname{Aut}(Q_{n-1})\) and a cyclic group of order 2.

MSC:

20N05 Loops, quasigroups
17A75 Composition algebras
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures

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