ABSTRACT
As pointed out in the chapters of focal-spreads, the only known focal-spreads are obtained from a construction of Beutelspacher [16]. In this chapter, the doubly transitive focal-spreads are completely determined as arising from Desarguesian spreads by k-cuts. The work in this chapter follows the article by the author and Montinaro [142], with appropriate modi…cations for this text. Since any collineation group of a focal-spread necessarily leaves in-
variant the focus L, it then becomes a natural question if a doubly transitive group acting on an a¢ ne focal-plane then forces the focalspread to have a similar structure as in the translation plane (i.e., t-spread) case. For example, is a doubly transitive a¢ ne focal-plane a k-cut of a semi…eld plane? First, consider simply a group G acting on a focal-spread of type
(t; k) over GF (q) that admits a primitive group acting on the partial Sperner k-spread. Since the degree is qt, by the O’Nan-Scott Theorem 7, either G = TG0 and T is an elementary Abelian group of order qt
(T is the socle of G), or G is almost simple. The proof that we cannot obtain the almost simple case uses very technical aspects of Kleidman and Liebeck [161] and Guralnick’s Theorem on simple groups of prime index, which provides us with a list of groups that can never apply in this situation. Although the proof is really beyond the scope of the text, we include it here but suggest that the reader have a copy of Kleidman and Liebeck in hand.
