# Local methods in fusion systems

## December 15-19, 2008

Organizer: Antonio Díaz Ramos

External Participants:

• Nadia Mazza (University of Lancaster)
• Sejong Park (Oberwolfach Institute)
• Radu Stancu (Universite de Picardie)

This is a workshop on fusion systems. There will be very few talks (1 per day) and plenty of time to interact. The interest is to make further progress in generalizing results from group theory to the fusion system setting. Nevertheless, the talks are not restricted to this topic, and they cover several topics related to fusion systems.

Programme

 Mon 15 13:15-14:15 Aud. 2 Antonio Díaz Introduction to fusion systems Tue 16 13:15-14:15 Aud. 2 Sejong Park Control of fusion and transfer Wed 17 10:15-11:15 Aud. 9 Nadia Mazza Oliver's conjecture Thu 18 13:15-14:15 Aud. 9 Radu Stancu Characteristic bisets and fusion Fri 19 10:15-11:15 Aud. 8 Adam Glesser Sparse fusion systems

Overview of the topics:

• Introduction to fusion systems: this will be an introduction to fusion systems theory with emphasis on the homotopy theoretical point of view. Main open problems of the theory will be described. Analogies with group theory will be discussed (recommended reading: [BLO04]).
• Sparse fusion systems: In studying fusion systems, one is inexorably led towards proving statements by considering a minimal counterexample and showing that the accompanying fusion system is constrained, hence modeled by a finite group. Commonly, the fusion system involved has very few subfusion systems. In an extreme case, we call the fusion system sparse. In this talk, we will give some basic properties of sparse fusion systems, allowing us to streamline the proof of a result of Kessar and Linckelmann as well as to give a proof of a new result based on an unpublished lemma of Navarro.
• Oliver's conjecture: In the proof of the Martino-Priddy conjecture in odd characteristic Bob Oliver defines a certain characteristic subgroup in any finite p-group (Definition 3.1 in [O04]). He then conjectures that this subgroup always contains the Thompson subgroup that is built up using the elementary abelian subgroups of maximal order (Conjecture 3.9 in [O04]). In fact, if it were true, then it would provide another proof of the Martino-Priddy conjecture in odd characteristic. But a proof of Oliver's conjecture is still awaiting, and only few cases have been shown. In this talk, we will present a survey of the conjecture and an update of the results known so far (recommended reading: [O04,GHL]).
• Control of fusion and transfer: We review the notion of control of fusion and transfer for finite groups and see how it can be generalized for (saturated) fusion systems.  Then we discuss recent generalizations of classical control of fusion/transfer theorems in local group theory to fusion systems - notably Glauberman-Thompson p-nilpotency theorem and Glauberman's ZJ- theorem [KL], Thompson's p-nilpotency theorem [DGMP1], Stellmacher's characteristic 2-subgroup theorem [OS], and Glauberman's K-group theorem [KL] [DGMP2].  Finally we discuss a new generalization of Yoshida's theorem on control of transfer to fusion systems.
• Characteristic bisets and fusion: To a saturated fusion system on a finite $p$-group $S$ Broto, Levi and Oliver associate a characteristic $S$-$S$ biset having  stability properties with respect to the fusion system. I'll explain that the existence of such characteristic biset implies the saturation of the fusion system. A characteristic biset associated to a saturated fusion system also satisfies some Frobenius reciprocity type properties (that I'll introduce and discuss). Again we have a converse statement saying that if a characteristic biset of a fusion system satisfies Frobenius reciprocity then the fusion system is saturated. This is joint work with Kari Ragnarsson.
• Generalizing results from group theory to fusion systems: Alperin's fusion theorem for finite groups is the most central result which has successfully been generalized to the fusion system setting. Other such results are Frobenius' theorem on normal p-complements, Glauberman and Thompson's p-nilpotency criterion and  Glauberman's ZJ-theorem.

We will use the visitors office 4.01 for informal meetings. The discussion topics in these meeting will be all of the above and possibly:

• Yoshida's theorem
• Category of elementary abelian p-subgroups
• Burnside ring of fusion systems

Bibliography:

• [BLO04] C. Broto, R. Levi, R. Oliver, The theory of p-local groups: a survey, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory,  51--84, Contemp. Math., 346, Amer. Math. Soc., Providence, RI, 2004.
• [GHL] D.J. Green, L. Hethelyi, and M. Lilienthal, On Oliver's p-group conjecture, preprint.
• [DGMP1] A. Diaz, A. Glesser, N. Mazza, S. Park, Glauberman and Thompson's theorems for fusion systems, Proc. Amer. Math. Soc., 137 (2009), 495-503.
• [DGMP2] A. Diaz, A. Glesser, N. Mazza, S. Park, Control of transfer
and weak closure in fusion systems, preprint (2008), preprint.
• [KL] R. Kessar, M. Linckelmann, ZJ-theorem for fusion systems, Trans. Amer. Math. Soc., 360 (2008), 3093--3206.
• [O04] R. Oliver, Equivalences of classifying spaces completed at odd primes, Math. Proc. Cambridge Philos. Soc., vol. 137 (2004), number 2, 321-347.
• [OS] S. Onofrei, R. Stancu, A characteristic subgroup for fusion systems, preprint (2008), preprint.