Flag-transitive Steiner Designs

This monograph provides an excellent development of the existence and nonexistence
of flag-transitive and other symmetric Steiner t-designs. In particular, it develops a
complete classification of all flag-transitive Steiner t-designs for strength t at least three.
The topic is a beautiful mixture of algebra and combinatorics, and it impinges on
many applications areas. Of particular value is the material providing the necessary
background in group theory, incidence geometry, number theory, and combinatorial
design theory to support a complete exposition of the many results. These form the
focus of the first three chapters. Chapter 4 then develops results on symmetric actions
of groups on Steiner systems, and provides many helpful examples. Chapter 5 then
states the main existence result for flag-transitive Steiner systems, and places this in
the context of related existence results for highly symmetric actions. Chapters 6 through
10 fill in the details of the existence proof. Steiner quadruple systems are treated in
Chapter 6, while strength three in general is treated in Chapter 7. Chapters 8 and 9
then treat the cases of strengths four and five, respectively. Finally Chapter 10 provides
the proof that no flag-transitive Steiner 6-design exists.
The presentation is lucid and accessible. Indeed the author has done a first rate job of
presenting material that involves many deep ideas and a number of technical issues. At
the same time, the monograph indicates useful next steps to take in the research topic.
Zentralblatt Math - Charles J. Colbourn (Tempe)