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MIT Combinatorics Seminar

The classification of flag-transitive Steiner designs

Michael Huber  (University of Tuebingen, Germany)

Friday, September 24, 2004    4:15 pm    Room 2-338

ABSTRACT

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.