3-cocycle for a group action

Definition
Let $$G$$ be a group acting (on the left) on an abelian group $$A$$ via a homomorphism of groups $$\varphi:G \to \operatorname{Aut}(A)$$ where $$\operatorname{Aut}(A)$$ is the automorphism group of $$A$$.

Explicit definition
A 3-cocycle for the action is a function $$f:G \times G \times G \to A$$ satisfying the following for all $$g_1,g_2,g_3,g_4 \in G$$ (here, the $$g_i$$s are allowed to be equal):

$$\!\varphi(g_1)(f(g_2,g_3,g_4)) - f(g_1g_2,g_3,g_4) + f(g_1,g_2g_3,g_4) - f(g_1,g_2,g_3g_4) + f(g_1,g_2,g_3) = 0$$

If we suppress $$\varphi$$ and use $$\cdot$$ for the action, we can rewrite this as:

$$\!g_1 \cdot f(g_2,g_3,g_4) - f(g_1g_2,g_3,g_4) + f(g_1,g_2g_3,g_4) - f(g_1,g_2,g_3g_4) + f(g_1,g_2,g_3) = 0$$

or equivalently:

$$\! g_1 \cdot f(g_2,g_3,g_4) + f(g_1,g_2g_3,g_4) + f(g_1,g_2,g_3) = f(g_1g_2,g_3,g_4) + f(g_1,g_2,g_3g_4)$$

Note that a function $$f:G \times G \times \to A$$ (without any conditions) is sometimes termed a 3-cochain for the group action.

Definition as part of the general definition of cocycle
A 3-cocycle for a group action is a special case of a defining ingredient::cocycle for a group action, namely $$n = 3$$. 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
The set of 3-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 3-cocycles $$f:G \times G \to A$$ can be identified with the group of $$\mathbb{Z}G$$-module maps from $$K$$ to $$A$$.

Importance

 * See 3-cocycle for trivial group action