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Fusion systems in algebra and topology. (English) Zbl 1255.20001
London Mathematical Society Lecture Note Series 391. Cambridge: Cambridge University Press (ISBN 978-1-107-60100-0/pbk). vi, 320 p. (2011).
This book is an excellent basic reference on fusion systems and an introduction to the field, interesting particularly for students and young mathematicians. It grew out of a workshop on fusion systems at the University of Birmingham in 2007.
Part one of this book, Introduction to fusion systems, contains foundational material about fusion systems, properties, definitions, notations, concepts, results.
The second part, The local theory of fusion systems, is an introduction to the local theory of saturated fusion systems, models for constrained saturated fusion systems, theorems on normal subsystems, classifying simple groups and fusion systems.
The third part is on Fusion and Homotopy theory: linking systems and classifying spaces of finite groups, abstract fusion and linking systems, the orbit category and its applications, examples of exotic fusion systems, open problems.
In the fourth part the authors discuss the role of fusion systems in modular representation theory: algebras and $$G$$-algebras, $$p$$-permutation algebras and saturated fusion systems, fusion and structure, block fusion systems and normal subgroups, open problems.
The book also contains exercises. It closes with an Appendix which records some of the basic material on finite groups, useful for those who approach fusion systems from the point of view of representation theory or homotopy theory.

MSC:
 20-02 Research exposition (monographs, survey articles) pertaining to group theory 20C20 Modular representations and characters 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 55R40 Homology of classifying spaces and characteristic classes in algebraic topology 20C33 Representations of finite groups of Lie type 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20J15 Category of groups 55Q99 Homotopy groups 55P99 Homotopy theory