Flag-transitive Steiner Designs by Michael HuberFlag-transitive Steiner Designs by Michael Huber

Flag-transitive Steiner Designs

byMichael Huber

Paperback | February 19, 2009

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The monograph provides the first full discussion of flag-transitive Steiner designs. This is a central part of the study of highly symmetric combinatorial configurations at the interface of several mathematical disciplines, like finite or incidence geometry, group theory, combinatorics, coding theory, and cryptography. In a sufficiently self-contained and unified manner the classification of all flag-transitive Steiner designs is presented. This recent result settles interesting and challenging questions that have been object of research for more than 40 years. Its proof combines methods from finite group theory, incidence geometry, combinatorics, and number theory.The book contains a broad introduction to the topic, along with many illustrative examples. Moreover, a census of some of the most general results on highly symmetric Steiner designs is given in a survey chapter.The monograph is addressed to graduate students in mathematics and computer science as well as established researchers in design theory, finite or incidence geometry, coding theory, cryptography, algebraic combinatorics, and more generally, discrete mathematics.
Title:Flag-transitive Steiner DesignsFormat:PaperbackDimensions:125 pages, 24.4 × 17 × 0.01 inPublished:February 19, 2009Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034600011

ISBN - 13:9783034600019

Reviews

Table of Contents

Preface.- 1. Incidence Structures and Steiner Designs.- 2. Permutation Groups and Group Actions.- 3. Number Theoretical Tools.- 4. Highly Symmetric Steiner Designs.- 5. A Census of Highly Symmetric Steiner Designs.- 6. The Classification of Flag-transitive Steiner Quadruple Systems.- 7. The Classification of Flag-transitive Steiner 3-Designs.- 8. The Classification of Flag-transitive Steiner 4-Designs.- 9. The Classification of Flag-transitive Steiner 5-Designs.- 10. The Non-Existence of Flag-transitive Steiner 6-Designs.- References.- Index.

Editorial Reviews

This monograph provides an excellent development of the existence and nonexistenceof flag-transitive and other symmetric Steiner t-designs. In particular, it develops acomplete 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 onmany applications areas. Of particular value is the material providing the necessarybackground in group theory, incidence geometry, number theory, and combinatorialdesign theory to support a complete exposition of the many results. These form thefocus of the first three chapters. Chapter 4 then develops results on symmetric actionsof groups on Steiner systems, and provides many helpful examples. Chapter 5 thenstates the main existence result for flag-transitive Steiner systems, and places this inthe context of related existence results for highly symmetric actions. Chapters 6 through10 fill in the details of the existence proof. Steiner quadruple systems are treated inChapter 6, while strength three in general is treated in Chapter 7. Chapters 8 and 9then treat the cases of strengths four and five, respectively. Finally Chapter 10 providesthe proof that no flag-transitive Steiner 6-design exists.The presentation is lucid and accessible. Indeed the author has done a first rate job ofpresenting material that involves many deep ideas and a number of technical issues. Atthe same time, the monograph indicates useful next steps to take in the research topic.Zentralblatt Math - Charles J. Colbourn (Tempe)