The Socle of Automorphism Groups of Linear Spaces 1 ; 2 Mathematics Subject Classiication 20b25, 05b05

This note is part of a general programme to classify the automorphism groups of nite linear spaces. There have been a number of contributions to this programme including two recent surveys, 8, 3]. One of the most signiicant contributions was the classiication of ag-transitive linear spaces, 1]. Since then the eeort has been to classify the line-transitive examples. These fall naturally into two classes, those where the action on points is primitive and those where the action on points is imprimitive. As a result of 9] it is known that the second only occurs nitely many times for a given line size. Two recent contributions to this problem have been 5] and 11]. In particular we examine the socle of a point-primitive line-transitive automorphism group of a linear space. The main result is:-Theorem 1 Let ? be a line-transitive, point-primitive automorphism group of a linear space S. Then the socle of ? is either elementary Abelian or simple. Work is underway on the situation when the socle is an alternating group. It is hoped that this work and the analysis of the situation for sporadic groups will soon be complete. A Linear Space S is a set P of points, together with a set L of distinguished subsets called lines such that any two points lie on exactly one line. This paper will be concerned with linear spaces with an automorphism group which is transitive on the lines. This implies that every line has the same number of points and we shall call such a linear space a Regular Linear Space. Moreover, we shall also assume that P is nite and that jLj > 1.