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

## Statement

### In cocycle and coboundary language at the origin

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

• $c$ is a symmetric 2-cocycle for trivial group action, i.e., $c(g,h) = c(h,g)$ for all $g,h \in G$.
• $c$ is a cyclicity-preserving 2-cocycle for trivial group action, i.e., $c(g,h) = 0$ if $\langle g,h \rangle$ is a cyclic group.

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

### In group extensions language at the origin

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

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

Suppose $G$ is a finitely generated abelian group and $A$ is an abelian group. Suppose $c_1$ and $c_2$ are 2-cocycles for the action of $G$ on $A$ such that $\operatorname{Skew}(c_1) = \operatorname{Skew}(c_2)$. Then, if $c_1$ and $c_2$ are both cyclicity-preserving, we must have $c_1 = c_2$.