Cocycle for trivial group action

Definition

Suppose $G$ is a group and $A$ is an abelian group.

Definition in terms of cocycle for a group action

A $n$-cocycle for trivial group action is a $n$-cocycle for a group action of $G$ on $A$, where the action is trivial.

Explicit definition

A $n$-cocycle for trivial group action of $G$ on $A$ is a function $f:G^{n}\to A$ satisfying the following for all $(g_{1},g_{2},\dots ,g_{n+1})\in G^{n+1}$:

$\!f(g_{2},g_{3},\dots ,g_{n+1})+\left[\sum _{i=1}^{n-1}(-1)^{i}f(g_{1},g_{2},\dots ,g_{i}g_{i+1},\dots ,g_{n})\right]+(-1)^{n+1}f(g_{1},g_{2},\dots ,g_{n})=0$

Particular cases

Value of $n$	Condition for being a $n$-cocycle for trivial group action	Further information
1	$\!f(g_{2})-f(g_{1}g_{2})+f(g_{1})=0$, or $f(g_{1}g_{2})=f(g_{1})+f(g_{2})$	It becomes a homomorphism of groups from $G$ to $A$, and hence, from the abelianization of $G$ to $A$
2	$\!f(g_{2},g_{3})-f(g_{1}g_{2},g_{3})+f(g_{1},g_{2}g_{3})-f(g_{1},g_{2})=0$, or $\!f(g_{2},g_{3})+f(g_{1},g_{2}g_{3})=f(g_{1}g_{2},g_{3})+f(g_{1},g_{2})$.	2-cocycle for trivial group action