Algebraic methods in the theory of combinatorial designs

Description

I. INTRODUCTION ------------------------------------------------------------- 1 1 .1 . PRELIMINARY REMARKS -------------------------------------------------------- 1 1.2. BACKGROUND ------------- -------------------------------------------------------------------------- 3 1.2 . a. DESIGN THEORY ------------------------------------------------------------------------ 3 1.2.b . LINEAR ALGEBRA ---------------------------------------------------------------------- 6 1 .2 .c . PERMUTATION GROUPS --------------------------------------------------------------- 8 1.3. CONVENTIONS -------------------------------------------------------------------------------------- 11 II. THE INCIDENCE ALGEBRA -------------------------------------------------------- 13 11.1. MOTIVATION --------------------------------------------------------------------------------------- 13 11.2. THE FUNDAMENTAL THEOREM --------------------------------------------------------------- 15 11.3. APPLICATIONS OF THE FUNDAMENTAL THEOREM ---------------------------------- 23 11.3 .a. GENERALIZED FISHERS INEQUALITY ------------------------------------ 24 11.3 .b. A CONJECTURE OF E. S. KRAMER --------------------------------------- 26 11.3 .c. GENERALIZED CONNOR'S INEQUALITIES ------------------------------ 30 III. CLASSIFICATION OF HOMOGENEOUS TRANSITIVE S(3 , {4,6} ,20) SYSTEMS 38 111.1. INTRODUCTION ------------------------------------------------------------------------------- 38 111.2. THE CASE v = 20 -------------------------------------------------------------------------- 40 111.3. THE STRUCTURE OF THE AUTOMORPHISM GROUP ------------------------------ 46 111.3 .a. THE CASE |omega| = 2 ----------------------------------------------------------------- 47 111.3 .b. THE CASE |omega| = 4 ----------------------------------------------------------------- 62 111.3.C. THE CASE |omega| = 5 --------------------------------------------------------------- 119 111.3 .d. THE CASE |omega| = 10 ------------------------------------------------------------- 123 111.4. SUMMARY -------------------------------------------------------------------------------------- 125 REFERENCES ------------------------------------------------------------------------------------------------------- 127