groups and degree preserving cohomology operations are corresponding to coefficient homomorphism . This follows from the Universal coefficient theorem for cohomology and the (m-1)-connectedness of . Some interesting examples for cohomology operations are Steenrod Squares and Powers, when are finite cyclic groups. When studying those the importance of the cohomology of with coefficients in becomes apparent quickly; some extensive tabeles of those groups can be found here. Group (co)homology One can define the group (co)homology of G with coefficients in the group A as the singular (co)homology of the Eilenberg-MacLane space with coefficients in A. Further Applications The loop space construction described above is used in string theory to obtain, for example, the string group, the fivebrane group and so on, as the Whitehead tower arising from the short exact sequence with the string group, and the spin group. The relevance of lies in the fact that there are the homotopy equivalences for the classifying space , and the fact . Notice that because the complex spin group is a group extension , the String group can be thought of as a "higher" complex spin group extension, in the sense of higher group theory since the space is an example of a higher group. It can be thought of the topological realization of the groupoid whose object is a single point and whose morphisms are the group . Because of these homotopical properties, the construction generalizes: any given space can be used to start a short exact sequence that kills the homotopy group in a topological group. See also Classifying space, for the case Brown representability theorem, regarding representation spaces Moore space, the homology analogue. Notes References Foundational articles Cartan seminar and applications The Cartan seminar contains many fundamental results about Eilenberg-Maclane spaces including their homology and cohomology, and applications for calculating the homotopy groups of spheres. http://www.numdam.org/volume/SHC_1954-1955__7/ Computing integral cohomology rings Derived functors of the divided power functors Integral Cohomology of Finite Postnikov Towers (Co)homology of the Eilenberg-MacLane spaces K(G,n) Other encyclopedic references Encyclopedia of Mathematics Homotopy theory