Automorphism groups of Cayley-Dickson loops. (Q441052)

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 \(\Aut(Q_n)\) is a direct product of \(\Aut(Q_{n-1})\) and a cyclic group of order 2.