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). Inaddition, especiallyfor? nitestructures, importantapplications to practical topics such as design theory, coding theory and cryptography have made the ? eld even more attractive. The subject matter of this research monograph is the study and class- cation of ? ag-transitive Steiner designs, that is, combinatorial t-(v, k, 1) designs which admit a group of automorphisms acting transitively on incident point-block pairs. As a consequence of the classi? cation of the ? nite simple groups, it has been possible in recent years to characterize Steiner t-designs, mainly for t=2, adm- ting groups of automorphisms with su? ciently strong symmetry properties. For Steiner 2-designs, arguably the most general results have been the classi? cation of all point 2-transitive Steiner 2-designs in 1985 by W. M. Kantor, and the almost complete determination of all ? ag-transitive Steiner 2-designs announced in 1990 byF. Buekenhout, A. Delandtsheer, J. Doyen, P. B. Kleidman, M. W. Liebeck, and J. Saxl. However, despite the classi? cation of the ? nite simple groups, for Steiner t-designs witht> 2 most of the characterizations of these types have remained long-standing challenging problems. Speci? cally, the determination of all ? - transitive Steiner t-designs with 3? t? 6 has been of particular interest and object of research for more than 40 years.
Inhaltsverzeichnis
Incidence Structures and Steiner Designs. - Permutation Groups and Group Actions. - Number Theoretical Tools. - Highly Symmetric Steiner Designs. - A Census of Highly Symmetric Steiner Designs. - The Classification of Flag-transitive Steiner Quadruple Systems. - The Classification of Flag-transitive Steiner 3-Designs. - The Classification of Flag-transitive Steiner 4-Designs. - The Classification of Flag-transitive Steiner 5-Designs. - The Non-Existence of Flag-transitive Steiner 6-Designs.
