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.

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.

Reviewer: Corina Mohorianu (Iaşi)

##### 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 |