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) $.



Non-degeneracy of product of group pairings
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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.


