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Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ arises as the skew $\kappa/\kappa^T$ of a 2-cocycle $\kappa \in Z^2(G,\mathbb{C}^\times)$, where $\kappa/\kappa^T$ depends only on the cohomology class of $\kappa$.

Is there a similar statement for symmetric bihomomorphisms? I'm looking for something of the form "Every symmetric bihomomorphism on $G$ is of the form $\kappa \cdot \kappa^T$ for a 2-cochain $\kappa$ with property X and two 2-cochains with property X give the same bihomomorphism if they differ by Y"

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  • $\begingroup$ What is $\kappa^T$? $\endgroup$
    – Mikhail Borovoi
    Commented May 25, 2017 at 16:25
  • $\begingroup$ the transpose, $\kappa^T(x,y)=\kappa(y,x)$ $\endgroup$
    – Bipolar Minds
    Commented May 25, 2017 at 18:05
  • $\begingroup$ And what is $Z^2(G)$? Do you mean $Z^2(G,\mathbb{C}^\times)$ ? $\endgroup$
    – Mikhail Borovoi
    Commented May 25, 2017 at 21:02
  • $\begingroup$ yes, changed it! $\endgroup$
    – Bipolar Minds
    Commented May 29, 2017 at 10:50

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