Fusion systems with some sporadic J-components

The research of the first author was partially supported by NSA Young Investigator Grant H98230-14-1-0312 and was supported by an AMS-Simons grant which allowed for travel related to this work.Peer reviewedPostprin

Punctured groups for exotic fusion systems

Acknowledgements. It was Andy Chermak who first asked the question in 2011 (arising out of his proof of existence and uniqueness of linking systems) of which exotic systems have localities on the set of non identity subgroups of a Sylow group. We thank him for comments on an earlier version of this paper and many helpful conversations, including during a visit to Rutgers in 2014, where he and the third author discussed the possibility of constructing punctured groups for the Benson-Solomon systems. We are grateful to George Glauberman for communicating to us several comments, corrections, and suggestions for improvement. We are especially grateful to the referee for pointing out that Lemma 4.8 conflicts with a lemma of Levi and Oliver, for alerting us to errors too numerous to mention, and for other suggestions. We would like to thank the Centre for Symmetry and Deformation at the University of Copenhagen for supporting a visit of the third named author, where some of the early work on this paper took place. Finally, the authors thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Groups, representations and applications”, where work on this paper was undertaken and supported by EPSRC grant no EP/R014604/1.Peer reviewedPublisher PD

Weight conjectures for fusion systems

Many of the conjectures of current interest in the representation theory of finite groups in characteristic pare local-to-global statements, in that they predict consequencesfor the representations of a finite group G given data about the representations of the p-local subgroups of G. The local structure of a block of a group algebra is encoded in the fusion system of the block together with a compatible family of Kulshammer-Puig cohomology classes. Motivated by conjectures in block theory, we state and initiate investigation of a number of seemingly local conjectures for arbitrary triples (S,F,α) consisting of a saturated fusion system F on a finite p-group S and a compatible family α

Weak closure and Oliver's p-group conjecture

To date almost all verifications of Oliver's p-group conjecture have proceeded by verifying a stronger conjecture about weakly closed quadratic subgroups. We construct a group of order 3^n for n = 49 which refutes the weakly closed conjecture but satisfies Oliver's conjecture.Comment: 9 page

Rigid automorphisms of linking systems

There has been recent interest in the connections between automorphisms of fusion systems, of their associated linking systems, and of the finite groups realizing them (when such groups exist). Comparison between the automorphism groups of fusion systems and their associated linking systems has importance for various group-like constructions in fusion systems. I plan to discuss work with Glauberman describing the group of rigid automorphisms of a linking system, i.e. those which are the identity on the fusion system. Then I plan to discuss a cohomological obstruction theory, similar to the Broto-Levi-Oliver obstruction theory for the existence and uniqueness of linking systems, which is setup to begin to provide a general framework for understanding when it is possible to define a unique centralizer of a fusion subsystem.Non UBCUnreviewedAuthor affiliation: University of Louisiana at LafayetteResearche