# Thread: Kernel of a homomorphism

1. ## Kernel of a homomorphism

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!

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 $\displaystyle J_k$.
see that your function is a well-defined (additive) group homomorphism, although it's trivial! now: $\displaystyle \hat { \frac {a}{b}} \in \ker g_k \Longleftrightarrow \frac {ap^k}{b} \in \mathbb{Z} \Longleftrightarrow \frac{a}{b} \in \frac{1}{p^k}\mathbb{Z}.$ thus: $\displaystyle \ker g_k=\{\hat{\frac{n}{p^k}}: \ n \in \mathbb{Z} \}=<\hat{\frac{1}{p^k}}>=J_k. \ \Box$
did i have to mention that the $\displaystyle 0$ element of $\displaystyle \mathbb{Z}_{p^{\infty}}$ is the coset $\displaystyle \mathbb{Z}$? i think you know that ...