Let p be a prime and let $\displaystyle \mathbb {Z} (p^ \infty ) = \{ \hat { \frac {a}{b} } \in \frac { \mathbb {Q} }{ \mathbb {Z} } : a,b \in \mathbb {Z} \ , \ b=p^i \ , \ i \geq 0 \} $.

Prove that the only proper subgroups of $\displaystyle \mathbb {Z} (p^ \infty ) $ are the finite cylic groups $\displaystyle J_k = < \hat { \frac {1}{p^k} } > $

For each positive integer k, define a function $\displaystyle g_k: \mathbb {Z} (p^ \infty ) \rightarrow \mathbb {Z} (p^ \infty ) $ by $\displaystyle g_k( \hat { \frac {a}{b} })= \hat { \frac {ap^k}{b}} $.

Show that the kernel of the homomorphism $\displaystyle g_k$ is $\displaystyle J_k

$.

How should I proceed with this? Thank you!