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Math Help - Kernel of a homomorphism

  1. #1
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    Kernel of a homomorphism

    Let p be a prime and let  \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  \mathbb {Z} (p^ \infty ) are the finite cylic groups J_k = < \hat { \frac {1}{p^k} } >

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

    Show that the kernel of the homomorphism g_k is J_k<br />
.

    How should I proceed with this? Thank you!
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let p be a prime and let  \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  \mathbb {Z} (p^ \infty ) are the finite cylic groups J_k = < \hat { \frac {1}{p^k} } >
    i already answered this question at least twice! see here.


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

    Show that the kernel of the homomorphism g_k is J_k<br />
.
    see that your function is a well-defined (additive) group homomorphism, although it's trivial! now: \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: \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 0 element of \mathbb{Z}_{p^{\infty}} is the coset \mathbb{Z}? i think you know that ...
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