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

From Groupprops

## Definition

Suppose is a group and 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:

- It is the subgroup of the second cohomology group for the trivial action of on , comprising those cohomology classes that can be represented by a cyclicity-preserving 2-cocycle for trivial group action.
- 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 and is defined as .