first cohomology group is the quotient of the so-called crossed homomorphisms, i.e. maps (of sets) f : G → M satisfying f(ab) = f(a) + af(b) for all a, b in G, modulo the so-called principal crossed homomorphisms, i.e. maps f : G → M given by f(g) = gm−m for some fixed m ∈ M. This follows from the definition of cochains above. If the action of G on M is trivial, then the above boils down to H1(G,M) = Hom(G, M), the group of group homomorphisms G → M, since the crossed homomorphisms are then just ordinary homomorphisms and the coboundaries (i.e. the principal crossed homomorphisms) must have image identically zero: hence there is only the zero coboundary. On the other hand, consider the case of where denotes the non-trivial -structure on the additive group of integers, which sends a to -a for every ; and where we regard as the group . By considering all possible cases for the images of , it may be seen that crossed homomorphisms constitute all maps satisfying and for some arbitrary choice of integer t. Principal crossed homomorphisms must additionally satisfy for some integer m: hence every crossed homomorphism sending -1 to an even integer is principal, and therefore: with the group operation being pointwise addition: , noting that is the identity element. H 2 If M is a trivial G-module (i.e. the action of G on M is trivial), the second cohomology group H2(G,M) is in one-to-one correspondence with the set of central extensions of G by M (up to a natural equivalence relation). More generally, if the action of G on M is nontrivial, H2(G,M) classifies the isomorphism classes of all extensions of G by M, in which the action of G on E (by inner automorphisms), endows (the image of) M with an isomorphic G-module structure. In the example from the section on immediately above, as the only extension of by with the given nontrivial action is the infinite dihedral group, which is a split extension and so trivial inside the group. This is in fact the significance in group-theoretical terms of the unique non-trivial element of . An example of a second group cohomology group is the Brauer group: it is the cohomology of the absolute Galois group of a field k which acts on the invertible elements in a separable closure: See also . Basic examples