Browsing by Subject "Fusion systems"

Now showing 1 - 5 of 5

Open Access
Cohomology of infinite groups realizing fusion systems
(Springer, 2019-06) Gündoğan, Muhammed Said; Yalçın, Ergün
Given a fusion system FF defined on a p-group S, there exist infinite group models, constructed by Leary and Stancu, and Robinson, that realize FF. We study these models when FF is a fusion system of a finite group G and prove a theorem which relates the cohomology of an infinite group model ππ to the cohomology of the group G. We show that for the groups GL(n, 2), where n≥5n≥5, the cohomology of the infinite group obtained using the Robinson model is different than the cohomology of the fusion system. We also discuss the signalizer functors P→Θ(P)P→Θ(P) for infinite group models and obtain a long exact sequence for calculating the cohomology of a centric linking system with twisted coefficients.
Open Access
Cohomology of infinite groups realizing fusion systems
(2019-09) Gündoğan, Muhammed Said
Given a fusion system F defined on a p-group S, there exist infinite group models, constructed by Leary and Stancu, and Robinson, that realize F. We study these models when F is a fusion system of a finite group G. If the fusion system is given by a finite group, then it is known that the cohomology of the fusion system and the Fp-cohomology of the group are the same. However, this is not true in general when the group is infinite. For the fusion system F given by finite group G, the first main result gives a formula for the difference between the cohomology of an infinite group model realizing the fusion F and the cohomology of the fusion system. The second main result gives an infinite family of examples for which the cohomology of the infinite group obtained by using the Robinson model is different from the cohomology of the fusion system. The third main result gives a new method for the realizing fusion system of a finite group acting on a graph. We apply this method to the case where the group has p-rank 2, in which case the cohomology ring of the fusion system is isomorphic to the cohomology of the group.
Open Access
Higher limits over the fusion orbit category
(Elsevier, 2022-06-09) Yalçın, Ergün
The fusion orbit category F‾C(G) of a discrete group G over a collection C is the category whose objects are the subgroups H in C, and whose morphisms H→K are given by the G-maps G/H→G/K modulo the action of the centralizer group CG(H). We show that the higher limits over F‾C(G) can be computed using the hypercohomology spectral sequences coming from the Dwyer G-spaces for centralizer and normalizer decompositions for G. If G is the discrete group realizing a saturated fusion system F, then these hypercohomology spectral sequences give two spectral sequences that converge to the cohomology of the centric orbit category Oc(F). This allows us to apply our results to the sharpness problem for the subgroup decomposition of a p-local finite group. We prove that the subgroup decomposition for every p-local finite group is sharp (over F-centric subgroups) if it is sharp for every p-local finite group with nontrivial center. We also show that for every p-local finite group (S,F,L), the subgroup decomposition is sharp if and only if the normalizer decomposition is sharp.
Open Access
An observation on the module structure of block algebras
(Taylor & Francis, 2019-05-26) Gelvin, Matthew J. K.
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 closed 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.
Open Access
On the basis of the burnside ring of a fusion system
(Elsevier, 2015) Gelvin, M.; Reeh, S.P.; Yalçın, E.
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 {αP}. We give an explicit formula that describes αP 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. © 2014 Elsevier Inc.