Abstract. We show that every saturated fusion system F has a unique minimal Fcharacteristic biset ΛF . We examine the relationship of ΛF with other concepts in plocal finite group theory: In the case of a constrained fusion system, the model for the fusion system is the minimal F-characteristic biset, and more generally, any centric linking system can be identified with the F-centric part of ΛF as bisets. We explore the grouplike properties of ΛF , and conjecture an identification of normalizer subsystems of F with subbisets of ΛF .
We consider the Burnside ring A(F ) of F -stable S-sets for a saturated fusion system F defined on a p-group S. It is shown by S.P. Reeh that the monoid of F -stable sets is a free commutative monoid with canonical basis {α P }. We give an explicit formula that describes α P as an S-set. In the formula we use a combinatorial concept called broken chains which we introduce to understand inverses of modified Möbius functions.
Let B be a p-block of the finite group G. We observe that the p-fusion of G constrains the module structure of B: Any basis of B that is invariant under the left and right multiplications of a chosen Sylow p-subgroup S of G must in fact form a semicharacteristic biset for the fusion system on S induced by G. The parameterization of such semicharacteristic bisets can then be applied to relate the module structure and defect theory of B. §0. Introduction. Let G be a finite group and S a Sylow p-subgroup of G. The left and right multiplicative actions of S on G give a partition of G by double cosets G = m i=1
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.