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

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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 second cohomology group for the trivial action of G on A, comprising those cohomology classes that can be represented by a 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).

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