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Thread: Kernel of a homomorphism

  1. #1
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    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!
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    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} } > $
    i already answered this question at least twice! see here.


    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 ...
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