Izbash V.I.
On automorphisms of the Cartesian square of a groupoid

In this paper, we study the automorphisms of the Cartesian square of a groupoid and give a new approach to describe them. We introduce the notion of a medial pair of groupoids and find a relation between such orthogonal pairs and automorphisms of Cartesian square of the given groupoid. For a groupoid with the identity element this relation is concretized. Then we obtain some results about automorphisms of the Cartesian square of a nonabelian simple group. Also we formulate two problems about medial pairs which are necessary to solve to describe automorphisms of the Cartesian square of a groupoid.