Flag-transitive Steiner Designs
The characterization of combinatorial or geometric structures in terms of their groups of automorphisms has attracted considerable interest in the last decades and is now commonly viewed as a natural generalization of Felix Klein’s Erlangen program(1872).
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Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît Perthame (Université Pierre et Marie Curie, Paris) Laurent Saloff-Coste (Cornell University, Ithaca) Igor Shparlinski (Macquarie University, New South Wales) Wolfgang Sprössig (TU Bergakademie Freiberg) Cédric Vilani (Ecole Normale Supérieure, Lyon)
Michael Huber
Flag-transitive
Steiner
Designs
Birkhäuser Verlag Basel . Boston . Berlin
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Contents Preface
vii
1
Incidence Structures and Steiner Designs 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Basic Properties and Fisher’s Inequality . . . . . . . . . . . . . . .
1 1 3 7
2
Permutation Groups and Group Actions 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Doubly Transitive Permutation Groups . . . . . . . . . . . . . . . .
11 11 13
3
Number Theoretical Tools 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Primitive Divisors and Zsigmondy’s Theorem . . . . . . . . . . . .
15 15 15
4
Highly Symmetric Steiner Designs 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Block’s Lemma and Related Results . . . . . . . . . . . . . . . . .
19 19 20 23
5
A Census of Highly Symmetric Steiner Designs 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Multiple Point-transitive Steiner Designs . . . . . . . . . . . . . . . 5.3 Flag-transitive Steiner Designs . . . . . . . . . . . . . . . . . . . .
27 27 27 29
6
The 6.1 6.2 6.3 6.4
35 35
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