The theory of 3-nets has grown from the needs of other branches of mathematics, such as quasigroup theory, differential geometry, foundations of geometry and finite geometries. 3-nets have been intensively studied since 1925
]). A powerful approach to the study of 3-nets comes from plane geometry: Collineations and projectivities can be defined in the same way as for an affine plane; then it is possible to develop a general theory for 3-nets both from Klein's and yon Staudt's points of view. Recently such a project has been realized by Barlotti and Strambach [6], see also
It would be of interest to develop this theory further for 3-nets of special kinds. In the present paper we study, from Klein's point of view, 3-nets co-ordinatized by commutative loops of exponent 2. We call them involutorial 3-nets with identitythe reason for this term will be given in Section 2. We deal with direction-preserving collineation groups which stabilize the transversal line et consisting of all points (x, x).
Concerning their permutation behaviour, we are interested in finite collineation groups acting on e t as a doubly transitive permutation group (or, equivalently, on the points off e t as a transitive permutation group). Our Theorem 5 states, that if such a collineation group contains a solvable normal subgroup then the underlying 3-net is 'classical', i.e. co-ordinatized by an abelian 2-group. The case when a minimal normal subgroup is simple appears to be complicated and also non-classical examples exist, see Section 3. However, it is natural to expect that this case would be successfully investigated by using the classification of the finite doubly transitive permutation groups.
Concerning collineations of particular kinds, Theorem 7 states a relationship between translations and associators.
As is known, to each commutative loop (Q, *) of exponent 2 there corresponds a 1-factorization of the complete graph on the vertex set Q. If (Q, *) has the inverse property (i.e. (Q, *) is a totally symmetric loop) then the corresponding 1-factorization is of Steiner-type, i.e. it comes from a Steiner