= Solution
Write the <finite abelian group> additively. A <character of a finite abelian group> is a <group homomorphism> $\chi:G\to\{z:|z|=1\}$. We first prove that there are exactly $|G|$ such <characters of a finite abelian group>, without assuming a structure theorem.
Use <extension of a character across a cyclic quotient>. Given a <subgroup> $H$ and $a\notin H$, let $k$ be the least positive integer with $ka\in H$. The <subgroup> $K=H+\langle a\rangle$ has $k$ cosets of $H$. If $\phi$ is a <character of a finite abelian group> on $H$, choose any of the $k$ roots $\lambda^k=\phi(ka)$ and define
$$
\widetilde\phi(h+ja)=\phi(h)\lambda^j.
$$
This is well-defined: two representations differ by an integer multiple of $ka$, and the root equation exactly cancels that difference. It is a <character of a finite abelian group>, and every extension arises from one of the $k$ choices of $\lambda$. Build a chain from the trivial <subgroup> to $G$ by adjoining elements. The <character of a finite abelian group> count multiplies by the same factor as the <subgroup> order at every step, so $|\widehat G|=|G|$.
For a nontrivial <character of a finite abelian group> $\eta$, choose $a$ with $\eta(a)\ne1$. Translating the group sum shows
$$
\sum_{x\in G}\eta(x)
=\eta(a)\sum_{x\in G}\eta(x),
$$
so the sum is zero. Applied to $\chi\overline\psi$, this proves
$$
\sum_{x\in G}\chi(x)\overline{\psi(x)}
=\begin{cases}|G|,&\chi=\psi,\\0,&\chi\ne\psi.\end{cases}
$$
The $|G|$ <orthogonal> nonzero <characters of a finite abelian group> therefore form a <basis> of all complex <functions> on $G$, a <vector space> of <dimension> $|G|$.
With the unnormalized <Fourier coefficients> $\widehat f(\chi)=\sum_{x\in G}f(x)\overline{\chi(x)}$, expansion in that basis gives the <Fourier inversion on a finite group>
$$
\boxed{f(x)=\frac1{|G|}\sum_{\chi\in\widehat G}\widehat f(\chi)\chi(x).}
$$
Under a normalized forward-transform convention the prefactor would instead be one. Thus the unspecified constant is determined by the convention, and the inversion itself follows directly from <character of a finite abelian group> counting and <orthogonality>.
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