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I am trying to learn about characters of finite abelian groups. A character is a homomorphism from a finite abelian group $G$ into the multiplicative group of complex numbers of absolute value 1. In my textbook (Finite Fields by Lidl and Niederreiter) there is the following question to which I am stuck:

Let $H$ be a subgroup of the finite abelian group $G$. Prove that the annihilator $A$ of H in $\widehat{G}$ (where $\widehat{G}$ is the group of characters of $G$) is isomorphic to $G/H$ and that $\widehat{G}/A$ is isomorphic to $H$.

This looks like something where the 1st isomorphism theorem for groups could be used, but I don't see how. Any ideas would be greatly appreciated.

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