On the Structure of Collineation Groups of Finite Projective Planes

into two parts: the determination first of the abstract structure of F, and then of the manner in which F acts on 3P. If F contains no perspectivities, almost nothing general is known about it. We shall study both questions inthe case where F is generated by involutory,perspectivities. To avoid unstructured situations, F will be assumed to contain at least two such perspectivities having different centres or axes. The determination of the structure of F is then not difficult if the order n of 2P is even (see §8 , Remark 1). Consequently, we shall concentrate on the case of odd n. The main reason for the difficulties occurring for odd nis that the permutation representation ofF o n the set ofcentres (o r axes) of involutory perspectivities has no nice group-theoretic properties; this is the exact opposite of the situation occurring for even n. Asusual, Z(T) and O(F) will denote the centre and largest normal subgroup of odd ,order of F; a are the centralizer of, and subgroup generated by, the subset A. THEOREM A. Let ^ be a finite projective plane of odd order n, and F a