Ellen Henke

Group cohomology and control of p-fusion

We show that if an inclusion of finite groups H≤G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories.The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2.

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

Control of fusion by Abelian subgroups of the hyperfocal subgroup

\mathcal{F}

PRODUCTS OF PARTIAL NORMAL SUBGROUPS

, there is a unique linking system whose objects are the subcentric subgroups of

A characterization of saturated fusion systems over abelian 2-groups

\mathcal{F}

Products in fusion systems

. Furthermore, the nerve of such a subcentric linking system is homotopy equivalent to the nerve of the centric linking system associated to

Minimal fusion systems with a unique maximal parabolic

\mathcal{F}

Direct and Central Products of Localities

. We believe that the existence of subcentric linking systems opens a new way for a classification of fusion systems of characteristic

Variety isomorphism in group cohomology and control of p-fusion

p

Fusion systems with Benson-Solomon components

-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.