Lenz-Barlotti I.4 Perspectivity Groups are Abelian

Abstract: We extend a 1972 result of Kantor and Pankin and give a new elementary proof of the assertion in the title for projective planes of arbitrary order. The main tool appears in the very rst book on group theory by Jordan in 1870. The Lenz-Barlotti classication of projective planes is based on the possible congurations of point-line pairs for which Desargues theorem holds. Desargues theorem holds for such a pair (p; l) if and only if the plane admits a full group of perspectivities with center p and axis l. By denition, such a perspectivity group xes all points on l, all lines on p and is maximally transitive consistent with these conditions. The Lenz-Barlotti gure of a projective plane is the set of point-line pairs for which is (p; l)-transitive and determines the Lenz-Barlotti class of . The plane is of class I.4 (respectively I.3) if its Lenz-Barlotti gure consists of the three non-incident point-line pairs of a triangle (respectively two of these pairs). These classes are two of the ve for which existence questions remain open [1].1 The purpose of this paper is to give a new elementary proof of: Theorem 1 Let be a projective plane of Lenz-Barlotti type I.4. Then its three transitive perspectivity groups are isomorphic and Abelian