Cohomological Methods in Group Theory by Ari Babakhanian

By Ari Babakhanian

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To this end, let e and f be two basis-maps for A, and let a and b be the corresponding 2-cocycles of G with values in l∗ , respectively. Then for σ ∈ G we have that eσ = φ(σ)fσ holds for some φ(σ) ∈ l∗ . This gives a map φ : G → l∗ defined by σ → φ(σ) that for each σ ∈ G satisfies the equality eσ = φ(σ)fσ . It is easy to check that for every σ, τ ∈ G the equality φ(σ)σ(φ(τ )) b(σ, τ ) = a(σ, τ ) φ(στ ) holds; hence, a and b are cohomologous 2-cocycles. 23 that (l, G, a) and (l, G, b) are isomorphic as k-algebras.

From this point forward we will assume the reader is familiar with the basics of group cohomology, for which we refer to Chapter 2 of [Mil11] or Chapter 4 of [CF67]. 21. Let k be a field, and let l ⊃ k be a finite Galois extension. Let a be a 2-cocycle of G with values in l∗ , and let A be the left vector space over L with basis {eσ }σ∈G for which multiplication is defined by ( ) ( ) ∑ ∑ ∑∑ xσ eσ · y τ eτ = xσ σ(yτ )a(σ, τ )eστ , σ∈G τ ∈G σ∈G τ ∈G where xσ , yτ ∈ L for σ, τ ∈ G. Then A is a central simple algebra over k that contains l as a strictly maximal subfield.

Then i=1 ⊗ D is k-algebra isomorphic to k ri=1 Di , where Di is a unique central division k-algebra up to isomorphism with ind(Di ) = pdi i and exp(Di ) = pei i for i = 1, . . , r. Proof. 6]. 28 Bibliography [AM69] M. Atiyah and I. G. Macdonald, Introduction to commutative algebra, AddisonWesley Publishing Company, 1969. [Ax64] J. Ax, Zeroes of polynomials over finite fields, American Journal of Mathematics 86 (1964), no. 2, 255–261. [Bou73] N. Bourbaki, Elements of mathematics - Algebra, Springer-Verlag, 1973.

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