crossed homomorphism
A crossed homomorphism is what an ordinary homomorphism becomes when the target is being shaken by a group action. An honest homomorphism satisfies f(gh) = f(g) + f(h); but if g also moves the elements of the target around, the naive rule must be corrected by inserting that movement. The corrected rule, f(gh) = f(g) + g·f(h), is the crossed homomorphism law, and it is exactly the condition for f to be a 1-cocycle.
Let G be a group and A a G-module. A crossed homomorphism (or 1-cocycle) is a function f : G -> A satisfying f(gh) = f(g) + g·f(h) for all g, h in G. The set of all such functions is the group Z^1(G, A) of 1-cocycles. A crossed homomorphism is called principal (a 1-coboundary) if it has the form f(g) = g·a - a for some fixed a in A; these form the subgroup B^1(G, A), and the quotient Z^1/B^1 is the first cohomology H^1(G, A).
There is a clean structural picture: crossed homomorphisms G -> A correspond bijectively to group-theoretic sections of the semidirect product A ⋊ G -> G, i.e. to subgroups complementary to A; principal ones correspond to conjugate complements. When G acts trivially the crossed law collapses to the ordinary one and crossed homomorphisms are just homomorphisms G -> A, recovering H^1(G, A) = Hom(G, A). In Galois theory crossed homomorphisms into L^* are the protagonists of Hilbert's Theorem 90.
For multiplicative G-modules one writes the law multiplicatively: f(gh) = f(g) · g(f(h)). Hilbert's Theorem 90 says every such crossed homomorphism into L^* is principal, i.e. of the form f(g) = g(b)/b for some b in L^*.