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Fusion Systems in Algebra and Topology

2011-08-25 Mathematics
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Release : 2011-08-25
Genre : Mathematics
Kind : eBook
Book Rating : 002/5 ( reviews)

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Book Synopsis Fusion Systems in Algebra and Topology by : Michael Aschbacher

Download or read book Fusion Systems in Algebra and Topology written by Michael Aschbacher. This book was released on 2011-08-25. Available in PDF, EPUB and Kindle. Book excerpt: A fusion system over a p-group S is a category whose objects form the set of all subgroups of S, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. This book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.

Fusion Systems in Algebra and Topology

2014-05-14 MATHEMATICS
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Release : 2014-05-14
Genre : MATHEMATICS
Kind : eBook
Book Rating : 844/5 ( reviews)

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Book Synopsis Fusion Systems in Algebra and Topology by : Michael Aschbacher

Download or read book Fusion Systems in Algebra and Topology written by Michael Aschbacher. This book was released on 2014-05-14. Available in PDF, EPUB and Kindle. Book excerpt: A fusion system over a p-group S is a category whose objects form the set of all subgroups of S, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. The book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.

The Theory of Fusion Systems

2011-06-23 Mathematics
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Release : 2011-06-23
Genre : Mathematics
Kind : eBook
Book Rating : 965/5 ( reviews)

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Book Synopsis The Theory of Fusion Systems by : David A. Craven

Download or read book The Theory of Fusion Systems written by David A. Craven. This book was released on 2011-06-23. Available in PDF, EPUB and Kindle. Book excerpt: The first book to deal comprehensively with the theory of fusion systems.

Fusion Action Systems

2010
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Release : 2010
Genre :
Kind : eBook
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Book Synopsis Fusion Action Systems by : Matthew Justin Karcher Gelvin

Download or read book Fusion Action Systems written by Matthew Justin Karcher Gelvin. This book was released on 2010. Available in PDF, EPUB and Kindle. Book excerpt: The study of fusion first arose in the local theory of finite groups. Puig abstracted the fusion data of a finite group to the notion of fusion system, an object that reflects local data in more abstract algebraic settings, such as the block theory of finite groups. Martino and Priddy conjectured that the algebraic data of a fusion system of a finite group should have a topological interpretation, which result was proved by Oliver using the notion of p-local finite group introduced by the team of Broto, Levi, and Oliver. The study of fusion systems and p-local finite groups thus provides a bridge between algebraic fields related to local group theory and algebraic topology. In this thesis we generalize the notion of abstract fusion system to model the local structure of a group action on a finite set. The resulting fusion action systems can be seen as a generalization of the notion of abstract fusion system, though we describe other possible interpretations as well. We also develop the notion of a p-local finite group action, which allows for connections between fusion action system theory and algebraic topology.

Rank 2 Amalgams and Fusion Systems

2024-04-22 Mathematics
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Release : 2024-04-22
Genre : Mathematics
Kind : eBook
Book Rating : 606/5 ( reviews)

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Book Synopsis Rank 2 Amalgams and Fusion Systems by : Martin van Beek

Download or read book Rank 2 Amalgams and Fusion Systems written by Martin van Beek. This book was released on 2024-04-22. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides a comprehensive treatment of the classification of small fusion systems, that is, fusion systems with few essential subgroups. It demonstrates a broad range of techniques from local group theory and fusion systems, several of which can be applied in more general settings. Addressing research problems that have not been treated in the past, it is the first text to explicitly use the amalgam method in this context. Fusion systems offer an enticing way to unify various p-local methods employed in group theory, representation theory and homotopy theory; but as abstract constructions they are still somewhat mysterious. This book paves the way to a broad and systematic study of these categories by applying the amalgam method, thus modernizing a methodology widely used to understand the local structure of finite groups. With this comes an introduction to several vital techniques in local group theory, a generous survey of the structure and modular representation theory of some important families of finite groups, and a demonstration of the value of combinatorial methods in finite group theory and fusion systems. Primarily aimed at researchers active in fusion systems and group amalgams, the book will also be of interest to anyone working with finite groups and their modular representations, group actions on trees, or classifying spaces. The inclusion of preliminary chapters outlining the theoretical prerequisites make it ideal for a short lecture course or as a reading group text for early career researchers and graduate students.

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