the group homology of GL(R). In some cases, stability results assert that the sequence of cohomology groups becomes stationary for large enough n, hence reducing the computation of the cohomology of the infinite general linear group to the one of some . Such results have been established when R is a field or for rings of integers in a number field. The phenomenon that group homology of a series of groups stabilizes is referred to as homological stability. In addition to the case just mentioned, this applies to various other groups such as symmetric groups or mapping class groups. Projective representations and group extensions In quantum mechanics we often have systems with a symmetry group We expect an action of on the Hilbert space by unitary matrices We might expect but the rules of quantum mechanics only require where is a phase. This projective representation of can also be thought of as a conventional representation of a group extension of by as described by the exact sequence Requiring associativity leads to which we recognise as the statement that i.e. that is a cocycle taking values in We can ask whether we can eliminate the phases by redefining which changes This we recognise as shifting by a coboundary The distinct projective representations are therefore classified by Note that if we allow the phases themselves to be acted on by the group (for example, time reversal would complex-conjugate the phase), then the first term in each of the coboundary operations will have a acting on it as in the general definitions of coboundary in the previous sections. For example, Extensions Cohomology of topological groups Given a topological group G, i.e., a group equipped with a topology such that product and inverse are continuous maps, it is natural to consider continuous G-modules, i.e., requiring that the action is a continuous map. For such modules, one can again consider the derived functor of . A special case occurring in algebra and number theory is when G is profinite, for example the absolute Galois group of a field. The resulting cohomology is called Galois cohomology. Non-abelian group cohomology Using the G-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group G with coefficients in a non-abelian group. Specifically, a G-group is a (not necessarily abelian) group A together with an action by G. The zeroth cohomology of G with