## Masterclass and workshop on

# Fusion systems and p-local group theory

## July 18-22, 2011.

### Background:

Traditionally, local group theory studies the relation between the global structure of a finite group and the structure of its (proper) subgroups. In particular, it investigates a group by looking at its p-local subgroups, i.e at the normalizers of its non-trivial p-subgroups. The successes of this field culminated in the classification of finite simple groups (in short: CFSG), but in the last decade, its methods have also found important new applications in the study of saturated fusion systems. Fusion systems were developed by Puig, mainly for the purposes of block theory. They can be seen as a generalization of the p-local structure of a finite group. Over the last years, the methods used in the CFSG were applied to work towards a classification of simple fusion systems.

Fusion systems are also of interest in homotopy theory. Broto, Levi and Oliver introduced the notion of a centric linking system associated to a fusion system in order to be able to talk about its classifying space. However, for a long time, it remained an open question whether there exists a unique centric linking system associated to each fusion system, although Oliver proved this in the group case using the CFSG. Moreover, Broto, Levi and Oliver developed an obstruction theory for the existence and uniqueness of centric linking systems. It had been conjectured that these obstructions vanish, or equivalently, that there exists a unique centric linking system associated to each saturated fusion system. Recently, this conjecture was proven by Chermak. An important ingredient in Chermak's proof is a purely group theoretic result, the general FF-module theorem of Meierfrankenfeld and Stellmacher, which relies on the CFSG. Chermak introduces in his proof also some new concepts that seem to be of interest on their own, namely objective partial groups and a new notion of localities. Oliver gave a reformulation of Chermak's proof using obstruction theory rather than localities. However, the proof still requires the group theoretic FF-module results.

### Main topics of the course:

- Motivation to study fusion systems and centric linking systems from the point of view of homotopy theory.
- Chermak's proof for the existence and uniqueness of centric linking systems using objective partial groups and localities.
- Oliver's reformulation of Chermak's proof through obstruction theory.
- Action of finite groups on GF(p)-modules, in particular FF-modules.
- The classification of finite simple groups (mainly of characteristic p -type) and of fusion systems.

**Venue:** Centre for Symmetry and Deformation, KU .

**Format of the course: **Three lecture series by Chermak, Oliver, and Stroth. Additional talks/lectures by Carles Broto, Ran Levi, Markus Linckelmann, Justin Lynd, Kári Ragarsson and Joana Ventura. The programm is supplemented by exercise sessions.

A detailed program is available here. Abstracts for all the talks and lectures can be found here.

**Exercises:**

- 1 Here is a sheet with preparatory exercises.
- 2 Here are the exercises that will be discussed in the exercise class on Monday.
- 3 Here are the exercises that will be discussed in Gernot Stroth's exercise class on Wednesday.
- 4 Here are exercises on obstruction theory that will be discussed in Matthew Gelvin's exercise class on Thursday.

**Lecture notes:**

- Handwritten notes from Bob Oliver's first lecture BO, his second lecture BO and his third lecture BO.
- Handwritten notes from Andrew Chermak's first two lectures AC and from his third lecture AC.
- Slides from Gernot Stroth's first lecture GS, his second lecture GS and his third lecture GS, handwritten notes GS from what he wrote on the blackboard.
- Handwritten notes from Ran Levi's Talk
- Handwritten notes from Joana Ventura's Talk
- Slides from Justin Lynd's Talk
- Handwritten notes from Markus Linckelmann's talk
- Handwritten notes from Kari Ragnarsson's talk
- Handwritten notes from Carles Broto's talk

**Practical Information: **All lectures and talks will be in Auditorium 8. Registration on Monday will also be in Auditorium 8 and starts at 8:30 am.

**Organizer:** Ellen Henke (Copenhagen)

**Participants:**

- Christine Bessenrodt (Leibniz Universität Hannover)
- Carles Broto (Universitat Autònoma de Barcelona)
- Andrew Chermak (Kansas State University)
- David Craven (University of Oxford)
- Antonio Diaz Ramos (Universidad de Málaga)
- Matthew Gelvin (University of Copenhagen)
- Adam Glesser (Suffolk University)
- Mathias Grimm (Universität Halle - Wittenberg)
- Jesper Grodal (University of Copenhagen)
- Ellen Henke (University of Copenhagen)
- Martin Wedel Jacobsen (University of Copenhagen)
- Radha Kessar (University of Aberdeen)
- Ran Levi (University of Aberdeen)
- Assaf Libman (University of Aberdeen)
- Markus Linckelmann (University of Aberdeen)
- Justin Lynd (Ohio State University)
- Jesper Michael Møller (University of Copenhagen)
- Toke Nørgård-Sørensen (University of Copenhagen)
- Bob Oliver (Université PARIS 13)
- Jørn Børling Olsson (University of Copenhagen)
- Sejong Park (Sogang University)
- Kári Ragnarsson (DePaul University, Chicago)
- Albert Ruiz Cirera (Universitat Autònoma de Barcelona)
- Nora Seeliger (Universität Regensburg)
- Jason Semeraro (University of Oxford)
- Eske Sparsø (University of Copenhagen)
- Radu Stancu (Université de Picardie)
- Gernot Stroth (Universität Halle - Wittenberg)
- Constantin-Cosmin Todea (Technical University of Cluj Napoca)
- Joana Ventura (Universidade Tecnica de Lisboa)

### Misc links and documents:

- M. Aschbacher, R. Kessar and Bob Oliver. Fusion systems in Algebra and Topology
*(preprint)* - A. Chermak. Fusion systems and localities
*(preprint, last updated June 1, 2011).* - B. Oliver. Existence and Uniqueness of Linking Systems: Chermak's proof via obstruction theory
*(preprint, last updated July 7, 2011)*. - Ulrich Meierfrankenfeld's homepage on finite groups of local characteristic p. Note in particular the preprint on the general FF-module theorem.