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 if $\displaystyle f: \mathbb {Z} (p^ \infty ) \rightarrow G $ is a surjective group homomorphism and G is not the trivial group $\displaystyle \{ 1_G \} $, then G is isomorphic to $\displaystyle \mathbb {Z} (p^ \infty ) $.