Cyclicity-preserving 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 cyclicity-preserving 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
cyclicity-preserving \! f(g,h) = 0 whenever \langle g,h \rangle is cyclic

The group of cyclicity-preserving 2-cocycles is denoted Z^2_{CP}(G,A).

Facts

Existence of a source group

For any group G, there exists an abelian group K such that for any abelian group A, the group of cyclicity-preserving 2-cocycles \! f:G \times G \to A can be identified with the group \operatorname{Hom}(K,A).

Further information: group of cyclicity-preserving 2-cocycles for trivial group action is naturally identified with group of homomorphisms from a particular abelian group

Examples

Extreme examples

Relation with other properties

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
IIP 2-cocycle for trivial group action 2-cocycle f such that \! f(g,1) = f(1,g) = f(g,g^{-1}) = 0 for all g \in G |FULL LIST, MORE INFO
normalized 2-cocycle for trivial group action 2-cocycle f such that \! f(g,1) = f(1,g) = 0 for all g \in G |FULL LIST, MORE INFO
2-cocycle for trivial group action IIP 2-cocycle for trivial group action|FULL LIST, MORE INFO