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

Contents

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:

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.

In group extensions language away from the origin

Proof

Proof in group extensions language at the origin

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