A Note on Serre’s Theorem in Group Cohomology
نویسنده
چکیده
In [7], Serre proves 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 H(G,Fp). For p = 2, this gives that (σG) = 0 where σG is the product of all nontrivial one dimensional classes in H(G,F2). In this note, we prove that if G is a nonabelian 2-group, then σG is also zero.
منابع مشابه
Digital cohomology groups of certain minimal surfaces
In this study, we compute simplicial cohomology groups with different coefficients of a connected sum of certain minimal simple surfaces by using the universal coefficient theorem for cohomology groups. The method used in this paper is a different way to compute digital cohomology groups of minimal simple surfaces. We also prove some theorems related to degree properties of a map on digital sph...
متن کاملDigital Borsuk-Ulam theorem
The aim of this paper is to compute a simplicial cohomology group of some specific digital images. Then we define ringand algebra structures of a digital cohomology with the cup product. Finally, we prove a special case of the Borsuk-Ulam theorem fordigital images.
متن کاملA FIXED POINT THEOREM AND RELATIVE ASPHERICITY by Max FORESTER and Colin ROURKE
We give short new geometric proofs of theorems of Bogley and Pride and of Serre. Both follow quickly from a global fixed point theorem. The Bogley–Pride theorem concerns aspherical relative group presentations and was applied in [5] to the multivariable adjunction problem. Serre’s theorem is a basic result concerning group cohomology and finite subgroups.
متن کاملOdd-dimensional Cohomology with Finite Coefficients and Roots of Unity
We prove that the triviality of the Galois action on the suitably twisted odd-dimensional étale cohomogy group with finite coefficients of an absolutely irreducible smooth projective variety implies the existence of certain primitive roots of unity in the field of definition of the variety. This text was inspired by an exercise in Serre’s Lectures on the Mordell–Weil theorem.
متن کاملA Finiteness Theorem in the Galois Cohomology of Algebraic Number Fields
In this note we show that if k is an algebraic number field with algebraic closure I and M is a finitely generated, free Zrmodule with continuous Ga\(k/k)action, then the continuous Galois cohomology group Hl(k, M) is a finitely generated Z,-module under certain conditions on M (see Theorem 1 below). Also, we present a simpler construction of a mapping due to S. Bloch which relates torsion alge...
متن کامل