Abstract. In his paper [6], S. Doro constructed a partial relationship between Moufang loops and groups with triality. We extend this relationship by showing that the following concepts are equivalent: Groups with triality and trivial centre, Moufang $3$-nets, Latin square designs in which every point is the centre of an automorphism, isotopy classes of Moufang loops. Using this new approach, we also give a simple proof to a theorem of Doro.

AMS Subject Classification (1991): 20N05

Received April 19, 1999, and in final form June 18, 2001. (Registered under 2809/2009.)