Group Extensions, Simple Algebras and Cohomology

Abstract

In this chapter we will study the structure of extensions of groups and the structure of central simple algebras over a field 𝔽. The theory of group extensions, $$N \triangleleft E \to G$$ , their existence and classification, will be reduced to two questions about low dimensional cohomology groups. Specifically, we will associate to G and the center C of N, abelian groups 2∅ (G; C) and 3∅ (G; C), depending only on G, G, and the action ∅ of G on C. The second group will contain an element, unambiguously defined for each triple (N, G, τ: G → Out(N)) so that → induces ∅ on restriction to C, and a necessary and sufficient condition for the existence of an extension with the data (N, G, τ) will be that that element be zero. Then, once it is known that some extension exists, the elements in the first group will count the number of distinct extensions which are possible up to an appropriate notion of isomorphism. These results are due to S. Eilenberg and S. MacLane, and first appeared in a paper in the Annals of Mathematics in 1947, [EM].KeywordsCohomology GroupCentral ExtensionDivision AlgebraSimple AlgebraGroup ExtensionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.