# Symmetric 2-cocycle for trivial group action

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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:
2-cocycle for a group action (particularly, 2-cocycle for trivial group action) $\! f(g_2,g_3) + f(g_1,g_2g_3) = f(g_1g_2,g_3) + f(g_1,g_2)$ for all $g_1,g_2,g_3 \in G$
symmetric $\! f(g,h) = f(h,g)$ for all $g,h \in G$