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 $\tilde{\chi }(h+jx)=\chi (h)z^j$ is well-defined and extends $\chi$ to $H+⟨x⟩$. 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.