Abstract:
In the last decades, there has been a great interest in classifying
Steiner t-designs with certain transitivity properties. For example,
W. M. Kantor characterized all point 2-transitive and F. Buekenhout et al.
all flag-transitive Steiner 2-designs. Both results depend on the
classification of the finite simple groups.
However, the classification of flag-transitive Steiner 3-designs is
a still open and longstanding problem and that of point 2-transitive seems
to be hopeless.
In this dissertation we use the classification of finite 2-transitive
permutation groups to classify all flag-transitive Steiner 3-designs with
small blocks.
In Chapter 1 we lay the foundations: combinatorial properties of designs, the
classification of the finite 2-transitive permutation groups and Zsigmondy's
theorem. In the following chapter, we prove that for Steiner t-designs with t
greater than 2, the flag-transitivity of the automorphism group implies its
point 2-transitivity. In Chapter 3, we classify all Steiner 3-designs with block
size 4, generalizing a result of H. Lüneburg. In Chapter 4, we finally give the
complete classifications of flag-transitive Steiner 3-Designs with block sizes
5,6 and 7.