= Extension of a unitary character from a finite abelian subgroup
Let $H\le G$ be finite abelian and let $\chi:H\to\mathbb T$ be a <character of a finite abelian group>. To adjoin $x\in G$, let $d$ be the least positive integer with $dx\in H$ and choose a complex number $z$ of modulus one with $z^d=\chi(dx)$. Then $\widetilde\chi(h+jx)=\chi(h)z^j$ is well-defined and extends $\chi$ to $H+\langle x\rangle$. There are exactly $d$ choices, corresponding to the $d$ roots. Iterating proves character extension to $G$, proves that $G$ has exactly $|G|$ characters, and proves separation of points by extending a primitive character of the cyclic subgroup generated by a nonzero point. No structure theorem is needed for this argument.
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