Symmetric 2-cocycle for trivial group action

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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_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