Homomorphism in each coordinate implies cocycle for trivial group action

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Statement

Suppose G is a group and A is an abelian group. Suppose f:G^n \to A is a function with the property that, for all i \in \{ 1,2,\dots,n \}, if we fix the entries in all coordinates but the i^{th} coordinate, the induced function from G to A is a homomorphism of groups.

Then, f is a n-cocycle for trivial group action.

Particular cases

n What being a homomorphism in each coordinate means What being a n-cocycle means Link to n-cocycle page
1 being a homomorphism of groups from G to A being a homomorphism of groups from G to A --
2 f:G \times G \to A satisfies, for all g_1,g_2,g_3 \in G, both of these: \! f(g_1g_2,g_3) = f(g_1,g_3) + f(g_2,g_3) and \! f(g_1,g_2g_3) = f(g_1,g_2) + f(g_1,g_3) f:G \times G \to A satisfies, for all g_1,g_2,g_3 \in G, the following: \! f(g_2,g_3) + f(g_1,g_2g_3) = f(g_1g_2,g_3) + f(g_1,g_2) 2-cocycle for trivial group action
3 f:G \times G \to A satisfies, for all g_1,g_2,g_3,g_4 \in G, all of these: \! f(g_1g_2,g_3,g_4) = f(g_1,g_3,g_4) + f(g_2,g_3,g_4),\! f(g_1,g_2g_3,g_4) = f(g_1,g_2,g_4) + f(g_1,g_3,g_4), \! f(g_1,g_2,g_3g_4) = f(g_1,g_2,g_3) + f(g_1,g_2,g_4) f:G \times G \to A satisfies, for all g_1,g_2,g_3,g_4 \in G, the following: \! f(g_2,g_3,g_4) + f(g_1,g_2g_3,g_4) + f(g_1,g_2,g_3) = f(g_1g_2,g_3,g_4) + f(g_1,g_2,g_3g_4) 3-cocycle for trivial group action