Flag-transitive collineation groups of nite projective spaces

An unpublished result of Perin [20] states that a subgroup of FL(n, q), n 23, which induces a primitive rank 3 group of even order on the set of points of PG(n 1, q), necessarily preset-es a symplectic polarity. (Such groups are known, if q # 2, by another theorem of Perin [19].) The present paper extends both Pcrin’s result and his method, in order to deal with some familiar problems concerning collineation groups of finite projective spaces; among these, 2transitive collineation groups [25], and the case q = 2 of Perin’s theorem [19]. An antifq is an ordered pair consisting of a hypcrplane and a point not on it; if the underlying s-ector space is endowed with a symplectic, unitary or orthogonal geometry, both the point and the pole of the hyperplane are assumed to be isotropic or singular. Our main results are the following four theorems.