On the Lenz-Barlotti Classification of Projective Planes

Theorem 1 implies that there does not exist any finite projective plane of Lenz type III . This completes results of Lff~V.BIYRG [11 and 12], C o F ~ [2], YAQUB [17], and HERr~G [6]. Theorem 2 implies that there does not exist any finite projective plane of Lenz-Barlotti type 1.6 or II.3. This result is due to YAQU~ [15 and 16], J6~sso~ [9], Lt~WBV~O [10], and CoFM~ [3]. Both theorems will be proved by means of some recent results on finite 2-transitive permutation groups due to S H ~ T [14] and Hv.Rr~G, Ka~TO~ and SErrz [7]. W e remark that, although induction was used in some of the above papers on planes of types I I I and 1.6, there is greater freedom to employ induction in the proofs of the theorems on permutation groups than in the purely geometric situations. One way of obtaining Theorems 1 and 2 from [14] and [7] is to consider the involutions in the permutation group induced on r by the group G generated by all the (X, XR)or (X, Xa)-perspeetivities. This approach requires the investigation of various special situations, and even and odd order planes must be handled differently. We have chosen to use a different approach which provides a more uniform proof.