I am interested in the computation of the Pontryagin duals of certain groups. At the moment I am stuck at a passage of the following book:
I.M. Gelfand, M.I. Graev, I.I. Pyatetskii-Shapiro, "Representation Theory and Automorphic Forms", W.B. Saunders Company, 1969
In the very first paragraph of Chapter 3, the dual of $\mathbb{Z}[p^{-1}]$ is described (the $p$-adic solenoid). I quote the conclusion of the computation:
Thus the character group of the additive group of all fractions of the form $\frac{a}{p^n}$ has the following structure. We take the additive group whose elements are the pairs $(\alpha,\beta)$, where $\alpha$ is a real number $\bmod 1$ and $\beta$ is a $p$-adic integer. In this group we factor out the subgroup of elements of the form $(r,r)$, where $r$ ranges over all rational numbers whose denominators are not divisible by $p$.
In my own interpretation and notation, the above suggests that the dual of $\mathbb{Z}[p^{-1}]$ is isomorphic to $(\mathbb{R}/ \mathbb{Z}\oplus\mathbb{Z}_p) / \mathbb{Z}_{(p)}$. I have two questions.
It is not clear to me where $\mathbb{Z}_{(p)}$ comes from during the computation (assuming it is correct).
It is well-known (from other sources) that the dual of $\mathbb{Z}[p^{-1}]$ is isomorphic to $(\mathbb{R}\oplus\mathbb{Z}_p) / \mathbb{Z}$. Are these two computations at odds with each other?
Any comments will be much appreciated.