# Symmetric and cyclicity-preserving 2-cocycle implies 2-coboundary

## Contents

## Statement

### In cocycle and coboundary language at the origin

Suppose is a finitely generated abelian group and is an abelian group. Suppose is a 2-cocycle for trivial group action of on satisfying the following two conditions:

- is a symmetric 2-cocycle for trivial group action, i.e., for all .
- is a cyclicity-preserving 2-cocycle for trivial group action, i.e., if is a cyclic group.

Then, is a 2-coboundary for trivial group action.

### In group extensions language at the origin

Suppose is an abelian group with a subgroup and quotient group . Suppose is finitely generated. Suppose there exists a 1-closed transversal of in (i.e., a collection of coset representatives, in other words, intersects each coset at exactly one point, with the property that any power of an element in is also in ). Then, in fact, is a direct factor of and in particular we can write as an internal direct product .

### In cocycle and coboundary language away from the origin

Suppose is a finitely generated abelian group and is an abelian group. Suppose and are 2-cocycles for the action of on such that . Then, if and are *both* cyclicity-preserving, we must have .

### In group extensions language away from the origin

## Proof

### Proof in group extensions language at the origin

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