12,936 research outputs found
Products in Fusion Systems
We revisit the notion of a product of a normal subsystem with a -subgroup
as defined by Aschbacher. In particular, we give a previously unknown, more
transparent construction.Comment: 18 pages; minor revisions; to appear in the Journal of Algebr
Products of partial normal subgroups
We show that the product of two partial normal subgroups of a locality (in
the sense of Chermak) is again a partial normal subgroup. This generalizes a
theorem of Chermak and fits into the context of building a local theory of
localities.Comment: 10 pages; compared to the first version some more background on
localities included; this version is accepted to the Robert Steinberg
Memorial Issue of the Pacific Journal of Mathematic
A characterization of saturated fusion systems over abelian 2-groups
Given a saturated fusion system over a -group , we prove
that is abelian provided any element of is -conjugate to
an element of . This generalizes a Theorem of Camina--Herzog, leading to
a significant simplification of its proof. More importantly, it follows that
any -block of a finite group has abelian defect groups if all
-subsections are major. Furthermore, every -block with a symmetric stable
center has abelian defect groups.Comment: 4 pages. One corollary added in the updated version. Accepted to
appear in Adv. Mat
Der Bewegungsbegriff in der neueren Geometrie und seine Adaption im elementaren Geometrieunterricht
Subcentric linking systems
Linking systems are crucial for studying the homotopy theory of fusion
systems, but are also of interest from an algebraic point of view. We propose a
definition of a linking system associated to a saturated fusion system which is
more general than the one currently in the literature and thus allows a more
flexible choice of objects of linking systems. More precisely, we define
subcentric subgroups of fusion systems in a way that every quasicentric
subgroup of a saturated fusion system is subcentric. Whereas the objects of
linking systems in the current definition are always quasicentric, the objects
of our linking systems only need to be subcentric. We prove that, associated to
each saturated fusion system , there is a unique linking system
whose objects are the subcentric subgroups of . Furthermore, the
nerve of such a subcentric linking system is homotopy equivalent to the nerve
of the centric linking system associated to . We believe that the
existence of subcentric linking systems opens a new way for a classification of
fusion systems of characteristic -type. The various results we prove about
subcentric subgroups give furthermore some evidence that the concept is of
interest for studying extensions of linking systems and fusion systems.Comment: 42 pages, accepted to Trans. Amer. Math. So
Centralizers of normal subgroups and the -Theorem
Glauberman's -theorem and analogous statements for odd primes show that,
for any prime and any finite group with Sylow -subgroup , the
centre of is determined by the fusion system
. Building on these results we show a statement that seems a
priori more general: For any normal subgroup of with
, the centralizer is expressed in terms of the
fusion system and its normal subsystem induced by .Comment: 3 pages; to appear in the Journal of Algebr
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