Pontryagin Duality

Question

A pro-p group is a profinite group in which every open normal subgroup has index equal to some power of p. Suppose G is pro-p Group. Then, Whether there is a natural bijection between $Hom_{cont}(G, \Bbb Q_p/Z_p)$ and $Hom_{cont}(G, \Bbb Q/Z) $? (where $Q_p$ and $Z_p $ denotes the p-adic rationals and integers respectively.)

Answer 1

• $(\Bbb{Q/Z})[p^\infty]= \Bbb{Z[p^{-1}]/Z}\cong \Bbb{Q_p/Z_p}$

• That $G$ is a pro $p$-group (with the profinite topology) means $\forall g, \lim_{k \to \infty} [p^k] g = 0$,

  Whatever Hausdorff topology we put on $\Bbb{Q/Z}$,

  For a continuous homomorphism $f : G \to\Bbb{Q/Z}$ for all $g\in G$ we must have $\lim_{k \to \infty} [p^k] f(g) = 0$

  ( $\lim_{k \to \infty} [p^k] f(g)$ is in the discrete topology of the finite group generated by $f(g)$)

  which means $[p^r] f(g)=0$ and $f$ takes its values in $(\Bbb{Q/Z})[p^\infty]$.