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Báo cáo toán học: " Line-transitive Automorphism Groups of Linear Spaces1"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Line-transitive Automorphism Groups of Linear Spaces1. | Line-transitive Automorphism Groups of Linear Spaces1 Alan R Camina and Susanne Mischke School of Mathematics University of East Anglia Norwich NR4 7TJ UK Submitted May 18 1995 Accepted December 21 1995 e-mail A.Camina@uea.ac.uk misch.ke_s@jpmorgan.com Abstract In this paper we prove the following theorem. Let S be a linear space. Assume that S has an automorphism group G which is line-transitive and point-imprimitive with k 9. Then S is one of the following - a A projective plane of order 4 or 7 a One of 2 linear spaces with v 91 and k 6 b One of 467 linear spaces with v 729 and k 8. In all cases the full automorphism group Aut S is known. 1 Introduction A linear space S is a set of points P together with a set of distinguished subsets L called lines such that any two points lie on exactly one line. This paper will be concerned with linear spaces in which every line has the same number of points and we shall call such a space a regular linear space. Moreover we shall also assume that P is finite and that L 1. The number of points will be denoted by v the number of lines by b the number of points on a line will be denoted by k and the number of lines through a point by r. We shall assume that k 2. Regular linear spaces are also called 2 v k 1 block designs and sometimes Steiner Systems. The choice of notation was determined by the use of the language of linear spaces by a number of authors as well as the need to study the fixed points of automorphisms. Such subsets inherit the structure of the linear space but not of the block design. mathematics Subject Classification 05B05 20C25 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 3 1996 R3 2 In this paper we investigate the properties of linear spaces which have an automorphism group which is transitive on lines. Clearly such a space is automatically a regular linear space.It follows from a result of Block 1 that a line-transitive automorphism group of a linear space is transitive on points. Recently Buekenhout .