# Cocycle for a group action

## Definition

Suppose $G$ is a group and $A$ is an abelian group, with an action $\varphi$ of $G$ on $A$. In other words, $\varphi$ is a homomorphism of groups from $G$ to $\operatorname{Aut}(A)$, the automorphism group of $A$.

### Definition in terms of bar resolution

A $n$-cocycle is an element in the $n^{th}$ cocycle group for the Hom complex from the bar resolution of $G$ to $A$, in the sense of $\mathbb{Z}G$-modules.

### Explicit definition

For $n$ a nonnegative integer, a $n$-cocycle for the action $\varphi$ of $G$ on $A$ is a function $f:G^n \to A$ such that, for all $g_1,g_2, \dots, g_{n+1} \in G$:

$\! \varphi(g_1)(f(g_2,g_3,\dots,g_{n+1})) + \left[ \sum_{i=1}^{n-1} (-1)^i f(g_1,g_2, \dots,g_ig_{i+1},\dots,g_n)\right] + (-1)^{n + 1}f(g_1,g_2,\dots,g_n) = 0$

If we suppress the symbol $\varphi$ and denote the action by $\cdot$, this becomes:

$\! g_1 \cdot f(g_2,g_3,\dots,g_{n+1})) + \left[ \sum_{i=1}^{n-1} (-1)^i f(g_1,g_2, \dots,g_ig_{i+1},\dots,g_n)\right] + (-1)^{n+1} f(g_1,g_2,\dots,g_n) = 0$

In particular, when the action is trivial, this is equivalent to saying that:

$\! f(g_2,g_3,\dots,g_{n+1}) + \left[ \sum_{i=1}^{n-1} (-1)^i f(g_1,g_2, \dots,g_ig_{i+1},\dots,g_n)\right] + (-1)^{n+1} f(g_1,g_2,\dots,g_n) = 0$

## Particular cases

$n$ Condition for being a $n$-cocycle Further information
1 For all $g_1,g_2\in G$, we have $\! g_1 \cdot f(g_2) - f(g_1g_2) + f(g_1) = 0$, equivalently $\! f(g_1g_2) = f(g_1) + g_1 \cdot f(g_2)$ 1-cocycle for a group action
2 For all $g_1,g_2,g_3\in G$, we have $\! g_1 \cdot f(g_2,g_3) - f(g_1g_2,g_3) + f(g_1,g_2g_3) - f(g_1,g_2) = 0$, equivalently $g_1 \cdot f(g_2,g_3) + f(g_1,g_2g_3) = f(g_1g_2,g_3) + f(g_2,g_3)$ 2-cocycle for a group action
3 For all $g_1,g_2,g_3,g_4 \in G$, we have $\!g_1 \cdot f(g_2,g_3,g_4) - f(g_1g_2,g_3,g_4) + f(g_1,g_2g_3,g_4) - f(g_1,g_2,g_3g_4) + f(g_1,g_2,g_3) = 0$, or equivalently, $\! g_1 \cdot f(g_2,g_3,g_4) + f(g_1,g_2g_3,g_4) + f(g_1,g_2,g_3) = f(g_1g_2,g_3,g_4) + f(g_1,g_2,g_3g_4)$ 3-cocycle for a group action