Characteristic bisets and local fusion subsystems

Abstract

Fusion systems are categories that contain the p-local structure of a finite group. Bisets are sets endowed with two coherent group actions. We investigate the relation between fusion systems and bisets in this thesis. Fusion systems that mimic the inclusion of a Sylow p-subgroup of a finite group are called saturated. Similarly, if S is a Sylow p-subgroup of G, then G regarded as an (S, S)-biset has special properties, which make it a characteristic biset for the p-fusion of G. These two concepts are linked in that a fusion system is saturated if and only if it has a characteristic biset. We give a proof for this result by following the work in [1] and [2]. Fusion systems have a notion of normalizer and centralizer subsystems, mimicking the notion for finite group theory. This thesis reviews a proof by Gelvin and Reeh [3] of a result of Puig [2] asserting that normalizer and centralizer fusion subsystems of a saturated fusion system are saturated. This result comes from the connection between saturation of fusion systems and the existence of characteristic bisets.