Overview

I1 More than one hundred years ago, Georg Frobenius [26] proved his remarkable theorem a?rming that, for a primep and a ?nite groupG, if the quotient of the normalizer by the centralizer of anyp-subgroup ofG is a p-group then, up to a normal subgroup of order prime top,G is ap-group. Ofcourse,itwouldbeananachronismtopretendthatFrobenius,when doing this theorem, was thinking the category - notedF in the sequel - G where the objects are thep-subgroups ofG and the morphisms are the group homomorphisms between them which are induced by theG-conjugation. Yet Frobenius' hypothesis is truly meaningful in this category. I2 Fifty years ago, John Thompson [57] built his seminal proof of the nilpotencyoftheso-called Frobeniuskernelofa FrobeniusgroupGwithar- ments - at that time completely new - which might be rewritten in terms ofF; indeed, some time later, following these kind of arguments, George G Glauberman [27] proved that, under some - rather strong - hypothesis onG, the normalizerNofasuitablenontrivial p-subgroup ofG controls fusion inG, which amounts to saying that the inclusionN?G induces an ? equivalence of categoriesF =F .

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From the reviews: ""The concept of Frobenius category (also called saturated fusion system) has been developed by Lluis Puig many years ago. ! This volume is a research monograph containing many original results which appear for the first time in print ! . The book ends with Appendix developing the author's point of view on cohomology of small categories. ! the book is a necessary tool for any researcher in the field."" (Andrei Marcus, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010) ""In recent years a new concept--called Frobenius category by Puig and fusion system by others--has become very successful in the representation theory of finite groups. ! the book gives an account of the early developments and definitions presented by Puig, as they were given in his first manuscripts. This certainly is of great importance because it makes the first, original definitions of Puig accessible to a wide audience."" (Alexander Zimmermann, Mathematical Reviews, Issue 2010 j)

"From the reviews: ""The concept of Frobenius category (also called saturated fusion system) has been developed by Lluis Puig many years ago. ! This volume is a research monograph containing many original results which appear for the first time in print ! . The book ends with Appendix developing the author's point of view on cohomology of small categories. ! the book is a necessary tool for any researcher in the field."" (Andrei Marcus, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010) ""In recent years a new concept--called Frobenius category by Puig and fusion system by others--has become very successful in the representation theory of finite groups. ! the book gives an account of the early developments and definitions presented by Puig, as they were given in his first manuscripts. This certainly is of great importance because it makes the first, original definitions of Puig accessible to a wide audience."" (Alexander Zimmermann, Mathematical Reviews, Issue 2010 j)"

From the reviews: The concept of Frobenius category (also called saturated fusion system) has been developed by Lluis Puig many years ago. ! This volume is a research monograph containing many original results which appear for the first time in print ! . The book ends with Appendix developing the author's point of view on cohomology of small categories. ! the book is a necessary tool for any researcher in the field. (Andrei Marcus, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010)

From the reviews: The concept of Frobenius category (also called saturated fusion system) has been developed by Lluis Puig many years ago. ! This volume is a research monograph containing many original results which appear for the first time in print ! . The book ends with Appendix developing the author's point of view on cohomology of small categories. ! the book is a necessary tool for any researcher in the field. (Andrei Marcus, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010) In recent years a new concept--called Frobenius category by Puig and fusion system by others--has become very successful in the representation theory of finite groups. ! the book gives an account of the early developments and definitions presented by Puig, as they were given in his first manuscripts. This certainly is of great importance because it makes the first, original definitions of Puig accessible to a wide audience. (Alexander Zimmermann, Mathematical Reviews, Issue 2010 j)

From the reviews: The concept of Frobenius category (also called saturated fusion system) has been developed by Lluis Puig many years ago. ... This volume is a research monograph containing many original results which appear for the first time in print ... . The book ends with Appendix developing the author's point of view on cohomology of small categories. ... the book is a necessary tool for any researcher in the field. (Andrei Marcus, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010) In recent years a new concept-called Frobenius category by Puig and fusion system by others-has become very successful in the representation theory of finite groups. ... the book gives an account of the early developments and definitions presented by Puig, as they were given in his first manuscripts. This certainly is of great importance because it makes the first, original definitions of Puig accessible to a wide audience. (Alexander Zimmermann, Mathematical Reviews, Issue 2010 j)

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