spaces. Recall that for a semi-direct product of groups there is an associated short exact sequence of groupsUsing the associated Eilenberg-Maclane spaces there is a Serre fibrationwhich can be put through a Serre spectral sequence. This gives an -pagewhich gives information about the group cohomology of from the group cohomology groups of . Note this formalism can be applied in a purely group-theoretic manner using the Lyndon–Hochschild–Serre spectral sequence. Cohomology of finite groups Higher cohomology groups are torsion The cohomology groups Hn(G, M) of finite groups G are all torsion for all n≥1. Indeed, by Maschke's theorem the category of representations of a finite group is semi-simple over any field of characteristic zero (or more generally, any field whose characteristic does not divide the order of the group), hence, viewing group cohomology as a derived functor in this abelian category, one obtains that it is zero. The other argument is that over a field of characteristic zero, the group algebra of a finite group is a direct sum of matrix algebras (possibly over division algebras which are extensions of the original field), while a matrix algebra is Morita equivalent to its base field and hence has trivial cohomology. If the order of G is invertible in a G-module M (for example, if M is a -vector space), the transfer map can be used to show that for A typical application of this fact is as follows: the long exact cohomology sequence of the short exact sequence (where all three groups have a trivial G-action) yields an isomorphism Tate cohomology Tate cohomology groups combine both homology and cohomology of a finite group G: where is induced by the norm map: Tate cohomology enjoys similar features, such as long exact sequences, product structures. An important application is in class field theory, see class formation. Tate cohomology of finite cyclic groups, is 2-periodic in the sense that there are isomorphisms A necessary and sufficient criterion for a d-periodic cohomology is that the only abelian subgroups of G are cyclic. For example, any semi-direct product has this property for coprime integers n and m. Applications Algebraic K-theory and homology of linear groups Algebraic K-theory is closely related to group cohomology: in Quillen's +-construction of K-theory, K-theory of a ring R is defined as the homotopy groups of a space Here is the infinite general linear group. The space has the same homology as i.e.,