Subgroups of group of characters of a finite abelian group

Question

Let G be a finite abelian group, H a subgroup of the group of characters of G. Is it true that H is the group of characters of some quotient group of G? Thanks for any help.

Answer 1

Let $\hat X = \operatorname{Hom}(X, S^1)$ denote the dual group of $X$. The map $f:G \to \hat H$ defined by $g \to \{\chi \to \chi(g)\}$ is surjective by Pontryagin duality. Hence $f$ induces an isomorphism $Q \to \hat H$ for some quotient $Q$ of $G$. Taking duality again gives $H = \hat Q$, as required.