Algebraic methods in the theory of combinatorial designs

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