Let $G$ be a finite abelian group, i.e., $G$ is a $\mathbb{Z}$-module. Consider the second component of the exterior algebra of $G$ over $\mathbb{Z}$, $\wedge^2 G=G\wedge G$. Suppose that $a:G\wedge G\rightarrow\mathbb{Q}/\mathbb{Z}$ is a nonzero, nondegenerate alternating form.
Does it follow that the exponent of the group $G$ and exponent of the image of $a$ is same?
Note that since $a$ is a nondegenerate form, the group $G$ is not cyclic.
Thank you for your time!