1-cocycle for a group action

Definition
A 1-cocycle is defined in the context of a group $$G$$, an abelian group $$A$$, and an action of $$G$$ on $$A$$ by automorphisms, i.e., a homomorphism of groups $$\varphi:G \to \operatorname{Aut}(A)$$ where $$\operatorname{Aut}(A)$$ is the automorphism group of $$A$$.

Explicit definition
A 1-cocycle of $$G$$ in $$A$$, also called a crossed homomorphism from $$G$$ to $$A$$, is a function $$f:G \to A$$ satisfying:

$$\! f(gh) = f(g) + \varphi(g)(f(h))$$

Here the group operation in $$G$$ is expressed multiplicatively, and the group operation in $$A$$ is expressed additively.

If we suppress $$\varphi$$ and simply denote the action by $$\cdot$$, the condition is:

$$\! f(gh) = f(g) + g \cdot f(h)$$

Definition as part of the general definition of cocycle
A 1-cocycle for a group action is a special case of a defining ingredient::cocycle for a group action in the case $$n = 1$$. This, in turn, is the notion of cocycle corresponding to the Hom complex from the defining ingredient::bar resolution of $$G$$ to $$A$$ as $$\mathbb{Z}G$$-modules.

Group structure
For a group $$G$$ acting on an abelian group $$A$$, the set of 2-cocycles for the action of $$G$$ on $$A$$ forms a group under pointwise addition.

As a group of homomorphisms
For any group $$G$$, we can construct a $$\mathbb{Z}G$$-module $$K$$ such that for any abelian group $$A$$, the group of 1-cocycles $$f:G \times G \to A$$ can be identified with the group of $$\mathbb{Z}G$$-module maps from $$K$$ to $$A$$.

Related notions

 * 1-cocycle for a Lie ring action

As stability automorphisms for an extension
Suppose we are given a group $$G$$ acting on an abelian group $$A$$, and a group $$E$$ which is the semidirect product of $$A$$ by $$G$$ with respect to this action. Now consider the group of those automorphisms of $$E$$ which fix $$A$$ pointwise and which are identity on $$G$$ (viewed as a quotient).

The claim is that this group is isomorphic to the group of 1-cocycles. The isomorphism is as follows. The automorphism takes every coset of $$A$$ to itself, and acts as identity on $$A$$. Hence it must simply translate each coset of $$A$$ by an element of $$A$$. Consider the function $$f:G \to A$$ by the map $$f(g) = g^{-1}\sigma(g)$$, in other words, this describes the amount by which each coset gets translated. The fact that $$\sigma$$ is an automorphism forces $$f$$ to be a 1-cocycle.

Interestingly, it is also true that the 1-coboundaries correspond to those stability automorphisms that arise as inner automorphisms by elements of $$A$$. Refer 1-coboundary for a group action.

The quotient of these groups (the first cohomology group) thus measures the group of stability automorphisms of $$1 \triangleleft A \triangleleft E$$ upto inner automorphisms of $$A$$.