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Cocycle for trivial group action

Contents

Definition

Suppose G is a group and A is an abelian group.

Definition in terms of cocycle for a group action

A n-cocycle for trivial group action is a n-cocycle for a group action of G on A, where the action is trivial.

Explicit definition

A n-cocycle for trivial group action of G on A is a function f:G^n \to A satisfying the following for all (g_1,g_2,\dots,g_{n+1}) \in G^{n+1}:

\! f(g_2,g_3,\dots,g_{n+1}) + \left[ \sum_{i=1}^{n-1} (-1)^i f(g_1,g_2, \dots,g_ig_{i+1},\dots,g_n)\right] + (-1)^{n+1} f(g_1,g_2,\dots,g_n) = 0

Particular cases

Value of n Condition for being a n-cocycle for trivial group action Further information
1 \! f(g_2) - f(g_1g_2) + f(g_1) = 0, or f(g_1g_2) = f(g_1) + f(g_2) It becomes a homomorphism of groups from G to A, and hence, from the abelianization of G to A
2 \! f(g_2,g_3) - f(g_1g_2,g_3) + f(g_1,g_2g_3) - f(g_1,g_2) = 0, or \! 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