On Divisible Designs and Twisted Field Planes Cinzia Cerroni, Antonino Giorgio Spera Dipartimento di Matematica ed Applicazioni, Universita Á di Palermo, via Archira® 34, I-90123 Palermo, Italy Received June 19, 1998; accepted November 13, 1998 Abstract: Starting from desarguesian and twisted ®eld planes we construct and study some classes of divisible designs admitting an automorphism group which is 2-transitive on the set of point classes. # 1999 John Wiley & Sons, Inc.J Combin Designs 7: 453±464, 1999 Keywords: divisible designs; R-permutation groups; division algebras; translation planes; automorphism groups: 05B30, 51E20, 51E15, 20B25, 20G40 1. INTRODUCTION A method to construct divisible designs (see the next section, [4] or [10] for de®nitions) by imprimitive permutation groups (G, X) with a special property was introduced in [17]. A group G with this special property admits an imprimitive block system R such that, if R is the equivalence relation associated with R, G is 2-R- homogeneous; that is, G is transitive on the nonordered pairs of non-R-equivalent elements of X. The divisible designs which are constructed by such a method admit a large automorphism group and often they have nice geometrical properties. Recently in [15] imprimitive groups which are 2-R-homogeneous were obtained from translation planes that have a 2-transitive collineation group on the line at in®nity l 1 . Such planes are desarguesian or Lu È neburg planes (see [14] and [8]), in the latter case one obtains interesting divisible designs admitting the Suzuki group as an automorphism group. In this article we consider translation planes which have a collineation group ®xing a point at in®nity and acting 2-transitively on the remaining points of l 1 . It is known (see [9]) that such planes are desarguesian or division algebra planes. Moreover, Correspondence to: Cinzia Cerroni, E-mail: Cerroni@ipamat.math.unipa.it # 1999 John Wiley & Sons, Inc. CCC 1063 8539/99/060453-12 453