Cyclicity-preserving subgroup of second cohomology group for trivial group action

Definition
Suppose $$G$$ is a group and $$A$$ is an abelian group. The group that we call the cyclicity-preserving subgroup of second cohomology group for trivial group action is defined in the following equivalent ways:


 * 1) It is the subgroup of the defining ingredient::second cohomology group for the trivial action of $$G$$ on $$A$$, comprising those cohomology classes that can be represented by a defining ingredient::cyclicity-preserving 2-cocycle for trivial group action.
 * 2) It is the quotient group of the group of cyclicity-preserving 2-cocycles for trivial group action by the group of cyclicity-preserving 2-coboundaries for trivial group action.

In symbols, this group is denoted $$H^2_{CP}(G,A)$$ and is defined as $$Z^2_{CP}(G,A)/B^2_{CP}(G,A)$$.

Related notions

 * IIP subgroup of second cohomology group for trival group action