Flag-transitive Steiner Designs

This monograph provides a full discussion of flag-transitive Steiner designs, a central part of the study of highly symmetric combinatorial configurations at the interface of several mathematical disciplines.

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.

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.