ABSTRACT
An ‘autotopism group’ of an affine translation plane pi is a group of collineations G that fixes two points on the line of infinity of the projective extension pi+ of pi and fixes an affine point P . The triangle so formed in pi+ is called the ‘autotopism’ triangle of G. Since the collineation group of pi is a semidirect product of the stabilizer of a point by the translation group, the affine point fixed by an autotopism group may always be taken arbitrarily. When autotopism groups become important for the analysis of a translation plane, there is a natural choice for the two infinite points fixed by the group. For example, if pi is a finite non-Desarguesian semifield plane of order pu, every collineation fixes the center of the affine elation group E of order pu, which we denote by (∞). Furthermore, the second fixed infinite point may be chosen arbitrarily since E is transitive on the set of remaining infinite points. More generally, we may coordinatize so that the two infinite points are (∞) and (0) and the affine fixed point is the zero vector (0, 0) of the associated underlying vector space.
