Browsing by Subject "Cohomology of groups"

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    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.
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    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.
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    Generalized Burnside rings and group cohomology
    (Academic Press, 2007-04-15) Hartmann, R.; Yalçın, E.
    We define the cohomological Burnside ring B n (G, M) of a finite group G with coefficients in a Z G-module M as the Grothendieck ring of the isomorphism classes of pairs [X, u] where X is a G-set and u is a cohomology class in a cohomology group H X n (G, M). The cohomology groups H X * (G, M) are defined in such a way that H X * (G, M) ≅ ⊕ i H * (H i, M) when X is the disjoint union of transitive G-sets G / H i. If A is an abelian group with trivial action, then B 1 (G, A) is the same as the monomial Burnside ring over A, and when M is taken as a G-monoid, then B 0 (G, M) is equal to the crossed Burnside ring B c (G, M). We discuss the generalizations of the ghost ring and the mark homomorphism and prove the fundamental theorem for cohomological Burnside rings. We also give an interpretation of B 2 (G, M) in terms of twisted group rings when M = k × is the unit group of a commutative ring. © 2006 Elsevier Inc. All rights reserved.
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    A note on Serre ' s theorem in group cohomology
    (American Mathematical Society, 2008) Yalçin, E.
    In 1987, Serre proved that if G is a p-group which is not elementary abelian, then a product of Bocksteins of one dimensional classes is zero in the mod p cohomology algebra of G, provided that the product includes at least one nontrivial class from each line in H1 (G,Fp). For p = 2, this gives that (σG) = 0, where σG is the product of all nontrivial one dimensional classes in H1 (G, F 2). In this note, we prove that if G is a nonabelian 2-group, then σG is also zero. © 2008 American Mathematical Society.
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    On nilpotent ideals in the cohomology ring of a finite group
    (2003) Pakianathan, J.; Yalçin, E.
    In this paper we find upper bounds for the nilpotency degree of some ideals in the cohomology ring of a finite group by studying fixed point free actions of the group on suitable spaces. The ideals we study are the kernels of restriction maps to certain collections of proper subgroups. We recover the Quillen-Venkov lemma and the Quillen F-injectivity theorem as corollaries, and discuss some generalizations and further applications.We then consider the essential cohomology conjecture, and show that it is related to group actions on connected graphs. We discuss an obstruction for constructing a fixed point free action of a group on a connected graph with zero "k-invariant" and study the class related to this obstruction. It turns out that this class is a "universal essential class" for the group and controls many questions about the groups essential cohomology and transfers from proper subgroups. © 2002 Elsevier Science Ltd. All rights reserved.