cohomology groups. Note could be replaced by any local system which is determined by a mapfor some abelian group . In the case of for letters, this is represented by a wedge sum of circles which can be showed using the Van-Kampen theorem, giving the group cohomology Group cohomology of an integral lattice For an integral lattice of rank (hence isomorphic to ), its group cohomology can be computed with relative ease. First, because , and has , which as abelian groups are isomorphic to , the group cohomology has the isomorphismwith the integral cohomology of a torus of rank . Properties In the following, let M be a G-module. Long exact sequence of cohomology In practice, one often computes the cohomology groups using the following fact: if is a short exact sequence of G-modules, then a long exact sequence is induced: The so-called connecting homomorphisms, can be described in terms of inhomogeneous cochains as follows. If is represented by an n-cocycle then is represented by where is an n-cochain "lifting" (i.e. is the composition of with the surjective map M → N). Functoriality Group cohomology depends contravariantly on the group G, in the following sense: if f : H → G is a group homomorphism, then we have a naturally induced morphism Hn(G, M) → Hn(H, M) (where in the latter, M is treated as an H-module via f). This map is called the restriction map. If the index of H in G is finite, there is also a map in the opposite direction, called transfer map, In degree 0, it is given by the map Given a morphism of G-modules M → N, one gets a morphism of cohomology groups in the Hn(G, M) → Hn(G, N). Products Similarly to other cohomology theories in topology and geometry, such as singular cohomology or de Rham cohomology, group cohomology enjoys a product structure: there is a natural map called cup product: for any two G-modules M and N. This yields a graded anti-commutative ring structure on where R is a ring such as or For a finite group G, the even part of this cohomology ring in characteristic p, carries a lot of information about the group the structure of G, for example the Krull dimension of this ring equals the maximal rank of an abelian subgroup . For example, let G be the group with two elements, under the