A Geometric Environment for Building up Loops

We start from the embedding of the Klein model of a hyperbolic plane H over a Euclidean field K in its direct motion group G := PSL_2 (K) and of both in PG(3, K). We present a geometric procedure to obtain loops which are related to suitable regular subsets of direct motions as transversals of the coset space G/D, where D is the subgroup of hyperbolic rotations fixing a given point o ∈ H. We investigate some properties of such loops and we determine their automorphism groups.