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.