Loop geometries

We introduce the construction of the semidirect product of a loop and its associate (or quasigroup) — the group uniquely generated by the loop. For a (left or right) loop the semidirect product is a group acting transitively on the loop so that the loop is provided with the structure of a homogeneous space, the stationary subgroup being its associate. The construction is reversible, viz.: any homogeneous space can be provided with the structure of a loop so that the semidirect product of it with the transassociate is isomorphic with the fundamental group of the homogeneous space and the transassociate is isomorphic with the stationarity group.