As a consequence of the classification of the finite
simple groups, it has been possible in recent years to
characterize Steiner t-designs, that is
t-(v,k,1) designs, mainly for the parameter t=2 with
sufficiently strong transitivity properties. Probably the most
general results have been the classification of all point
2-transitive Steiner 2-designs in 1985 by W. M. Kantor, and the almost
complete determination of all flag-transitive Steiner 2-designs announced in 1990
by F. Buekenhout, A. Delandtsheer, J. Doyen, P. B. Kleidman, M. W.
Liebeck and J. Saxl.
Nevertheless, for Steiner t-designs with
parameters t=3,4 such characterizations have remained
challenging open problems. In particular, the classifications of
all flag-transitive Steiner t-designs with t=3,4 are
known as long-standing and important problems.
In this talk, we shall give the complete classifications of all flag-transitive Steiner
t-designs with t=3,4. Our approach makes use of the classification of the finite
2-transitive permutation groups. The occurring examples and the most interesting
parts of the proofs shall be illustrated.