Normalized 2-cocycle for trivial group action

Definition

Suppose $G$ is a group and $A$ is an abelian group. A normalized 2-cocycle for trivial group action is a function $f:G\times G\to A$ that satisfies both these conditions:

• $f$ is a 2-cocycle for trivial group action: $f(g,hk)+f(h,k)=f(gh,k)+f(g,h)\mathrm{\forall }g,h,k,\in G$
• $f$ evaluates to the identity element of $A$ (written as 0 since $A$ is abelian) if either of its inputs is the identity element of $G$: $f(g,1)=f(1,g)=0\mathrm{\forall }g\in G$