Let $G$ be a finite abelian group. A quadratic form on $G$ is a map $q: G \to \mathbb{C}^*$ such that $q(q) = q(g^{-1})$ and the symmetric function $b(g,h):= \frac{q(gh)}{q(g)q(h)}$ is a bicharacter, i.e. $b(g_1g_2, h) = b(g_1, h)b(g_2, h)$ for all $g, g_1, g_2, h \in G.$

The quadratic form $q$ is called non-degenerate if the corresponding bicharacter $b$ is non-degenerate.

Question: Is there a non-degenerate quadratic form on every finite abelian group?

Motivation: it is used to make pointed braided/modular tensor categories, see Chapter 8 of this book (in particular Sections 8.4, 8.13 and 8.14).

Let $G$ be a finite abelian group. A quadratic form on $G$ is a map $q: G \to \mathbb{C}^*$ such that $q(g) = q(g^{-1})$ and the symmetric function $b(g,h):= \frac{q(gh)}{q(g)q(h)}$ is a bicharacter, i.e. $b(g_1g_2, h) = b(g_1, h)b(g_2, h)$ for all $g, g_1, g_2, h \in G.$

The quadratic form $q$ is called non-degenerate if the corresponding bicharacter $b$ is non-degenerate.

Question: Is there a non-degenerate quadratic form on every finite abelian group?

Motivation: it is used to make pointed braided/modular tensor categories, see Chapter 8 of this book (in particular Sections 8.4, 8.13 and 8.14).