Cyclicity-preserving 2-cocycle for trivial group action

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)	$\!f(g_{2},g_{3})+f(g_{1},g_{2}g_{3})=f(g_{1}g_{2},g_{3})+f(g_{1},g_{2})$ 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 $Z_{CP}^{2}(G,A)$.

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

• If $G$ is a cyclic group or a locally cyclic group, the group of cyclicity-preserving 2-cocycles is a trivial group.
• If $A$ is a trivial group, the group of cyclicity-preserving 2-cocycles is a trivial group.

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 $g\in G$			IIP 2-cocycle for trivial group action|FULL LIST, MORE INFO
2-cocycle for trivial group action				IIP 2-cocycle for trivial group action, Normalized 2-cocycle for trivial group action|FULL LIST, MORE INFO