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For $G$ finite abelian group, let $\eta,\omega:G \times G \to \mathbb{C}^\times$ be group pairings. What can I say about the (non-)degeneneracy of the product pairing $\eta \cdot \omega$ in terms of the (non-)degeneneracy of $\eta$ and $\omega$? The product is defined elementwise: $(\eta \cdot \omega) (g,h):= \eta(g,h) \cdot \omega(g,h) $.

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  • $\begingroup$ What is a group pairing? $\endgroup$ – Nick Gill Sep 21 '16 at 11:04
  • $\begingroup$ a map $\omega: G \times H \to \mathbb{C}^\times$, s.t. the induced map $G \to H^*$ is a group homomorphism, where $H^*$ is the dual group of $H$ $\endgroup$ – Bipolar Minds Sep 21 '16 at 11:16
  • $\begingroup$ If product is what I think it is, it's obviously nondegenerate of and only if so are $\mu$ and $\omega$. $\endgroup$ – Alex Degtyarev Sep 21 '16 at 16:50
  • $\begingroup$ the product is defined elementwise.. I dont see why this is obvious.. $\endgroup$ – Bipolar Minds Sep 21 '16 at 18:42
  • $\begingroup$ @AlexDegtyarev what kind of product did you think of? $\endgroup$ – Bipolar Minds Sep 21 '16 at 19:01
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This is not an answer, but rather a comment with too many characters to post as a comment.

If $A$ is a finite abelian group and $\chi\colon A\to A^*$ is an isomorphism, then $\chi$ determines a group isomorphism between $\langle \textrm{End}(A); +,-,0\rangle$ and the abelian group of pairings under pointwise product. Namely, if $\varphi\in\textrm{End}(A)$, then the corresponding pairing is $(a,b)\mapsto (\chi\circ \varphi(a))(b)$. Under this correpondence the property of nondegeneracy of a pairing corresponds to the property of injectivity of an endomorphism. Since $A$ is finite, nondegenerate pairings correspond to units in $\textrm{End}(A)$.

Hence the main question about pairings (What can I say about the nondegeneneracy of the product pairing $\eta\cdot\omega$ in terms of the nondegeneneracy of $\eta$ and $\omega$?) translates into the following question: When is the sum of two units in the ring $\textrm{End}(A)$ equal to a unit? Similarly, if you replace nondegeneracy with degeneracy in the original question, you should replace unit with nonunit in the reformulated question. Since $\textrm{End}(A)$ is a fairly typical finite ring, these questions are unlikely to have nontrivial answers.

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