coefficients in A is defined to be the subgroup of elements of A fixed by G. The first cohomology of G with coefficients in A is defined as 1-cocycles modulo an equivalence relation instead of by 1-coboundaries. The condition for a map to be a 1-cocycle is that and if there is an a in A such that . In general, is not a group when A is non-abelian. It instead has the structure of a pointed set – exactly the same situation arises in the 0th homotopy group, which for a general topological space is not a group but a pointed set. Note that a group is in particular a pointed set, with the identity element as distinguished point. Using explicit calculations, one still obtains a truncated long exact sequence in cohomology. Specifically, let be a short exact sequence of G-groups, then there is an exact sequence of pointed sets History and relation to other fields The low-dimensional cohomology of a group was classically studied in other guises, well before the notion of group cohomology was formulated in 1943–45. The first theorem of the subject can be identified as Hilbert's Theorem 90 in 1897; this was recast into Emmy Noether's equations in Galois theory (an appearance of cocycles for ). The idea of factor sets for the extension problem for groups (connected with ) arose in the work of Otto Hölder (1893), in Issai Schur's 1904 study of projective representations, in Otto Schreier's 1926 treatment, and in Richard Brauer's 1928 study of simple algebras and the Brauer group. A fuller discussion of this history may be found in . In 1941, while studying (which plays a special role in groups), Heinz Hopf discovered what is now called Hopf's integral homology formula , which is identical to Schur's formula for the Schur multiplier of a finite, finitely presented group: where and F is a free group. Hopf's result led to the independent discovery of group cohomology by several groups in 1943-45: Samuel Eilenberg and Saunders Mac Lane in the United States ; Hopf and Beno Eckmann in Switzerland; Hans Freudenthal in the Netherlands ; and Dmitry Faddeev in the Soviet Union (, ). The situation was chaotic because communication between these countries was difficult during World War II. From a topological point of view, the homology and cohomology of G was first defined as the homology and cohomology of a