Characteristic bisets and local fusion subsystems
Author
Tokmak, Mustafa Anıl
Advisor
Gelvin, Matthew Justin Karcher
Date
2018-09Publisher
Bilkent University
Language
en_US
Type
ThesisItem Usage Stats
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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.