View source for Symmetric and cyclicity-preserving 2-cocycle implies 2-coboundary

==Statement==

===In cocycle and coboundary language at the origin===

Suppose <math>G</math> is a [[finitely generated abelian group]] and <math>A</math> is an [[abelian group]]. Suppose <math>c</math> is a [[2-cocycle for trivial group action]] of <math>G</math> on <math>A</math> satisfying the following two conditions:

* <math>c</math> is a [[symmetric 2-cocycle for trivial group action]], i.e., <math>c(g,h) = c(h,g)</math> for all <math>g,h \in G</math>.
* <math>c</math> is a [[cyclicity-preserving 2-cocycle for trivial group action]], i.e., <math>c(g,h) = 0</math> if <math>\langle g,h \rangle</math> is a cyclic group.

Then, <math>c</math> is a [[2-coboundary for trivial group action]].

===In group extensions language at the origin===

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

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

Suppose <math>G</math> is a [[finitely generated abelian group]] and <math>A</math> is an [[abelian group]]. Suppose <math>c_1</math> and <math>c_2</math> are [[2-cocycle for trivial group action|2-cocycles]] for the action of <math>G</math> on <math>A</math> such that <math>\operatorname{Skew}(c_1) = \operatorname{Skew}(c_2)</math>. Then, if <math>c_1</math> and <math>c_2</math> are ''both'' [[cyclicity-preserving 2-cocycle for trivial group action|cyclicity-preserving]], we must have <math>c_1 = c_2</math>.

===In group extensions language away from the origin===

==Proof==

===Proof in group extensions language at the origin===

{{fillin}}