Determination of the automorphism group of a hyperbolic \(K\)-loop (Q1321694)

The second author was the first seeing that the point set \(H\) of a hyperbolic geometry can be provided with an addition (+) such that \((H,+)\) is a \(K\)-loop. \(K\)-loops are loops with the additional properties: (1) \(-(a + b) = (-a) + (-b)\) where \((-x) + x = 0\), (2) \(\delta_{a,b}(x) = (-(a+b)) + (a+ (b+x))\) is an automorphism of \((H,+)\), (3) \(\delta_{a,-a} = \text{id}\), (4) \(\delta_{a,b} = \delta_{a,b + a}\). \(K\)-loops which are derived from hyperbolic spaces are called hyperbolic \(K\)-loops. In this article the automorphism group \(\text{Aut}(H,+)\) of a hyperbolic \(K\)-loop is determined. It is shown that \(\text{Aut}(H,+)\) is the group of all similarities of the underlying hyperbolic space which fix the point 0.