On groups of Baer collineations acting on cartesian and translation planes

Abstract

Let G be a collineation group of a finite projective plane IZf, whose order is n. We shall call G a B-group if all its non-trivial elements are Baer collineations of n and (I GI, n) = 1. All known B-groups G are planar, i.e., their fixed elements form a subplane of I& which we shall always denote by &. If n, is a Baer subplane and n is a translation plane then Foulser [3] implies G is cyclic. Using this fact Ostrom [lOI showed that if S is a B-group of exponent 2, acting on a translation plane of order n, then S is planar and l7, has order ,I/lSl The object of this paper is to consider the problem of classifying those Bgroup G that act on the Cartesian and translation planes. Our main result, stated below, incorporates the theorems of Foulser and Ostrom mentioned above, and may be regarded as being a partial description of how B-groups act on translation planes. In particular, our result implies that G is planar and, more surprisingly, that if G is not of exponent 2, then n, is either a square root or a fourth root subplane of n.