Symmetric 2-cocycle for trivial group action

Definition

Suppose $G$ is a group and $A$ is an abelian group. A function $f:G\times G\to A$ is termed a symmetric 2-cocycle for trivial group action if it satisfies the following conditions:

Condition name	Expression for condition
2-cocycle for a group action (particularly, 2-cocycle for trivial group action)	$f(g_{2},g_{3})+f(g_{1},g_{2}g_{3})=f(g_{1}g_{2},g_{3})+f(g_{1},g_{2})$ for all $g_{1},g_{2},g_{3}\in G$
symmetric	$f(g,h)=f(h,g)$ for all $g,h\in G$