Symmetric and cyclicity-preserving 2-cocycle implies 2-coboundary
In cocycle and coboundary language at the origin
- 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 .