I am new to the concept of HSP. Previously, I saw how to solve hidden subgroup problem over $\mathbb{Z}_2^n$, which was Simon's algorithm. Over there the first step was to apply $H^{\otimes n}$, which was indeed the Fourier transform over $\mathbb{Z}_2^n$.
Now for any general finite abelian group, the course notes I have been following and in many other materials, the starting step is given as to form equal superposition over all the basis vectors i.e.,
$$\frac{1}{\sqrt{|G|}}\sum_{g\in G} | g,0\rangle$$
Here $G$ is the finite abelian group. The pair of register contains $|0,0\rangle$ at the beginning where the first register $|0\rangle$ means log$(|G|)$ qubits(Why?).
In case of Hadamard in Simon's algorithm it was easy to see this step, but in this case can anyone show me how do we get here? We have to apply Fourier transform I know, but the whole steps I am not getting.